text:
2.3: The Dot Product
- Last updated
- Save as PDF
- Page ID
- 32257
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Learning Objectives
- Calculate the dot product of two given vectors.
- Determine whether two given vectors are perpendicular.
- Find the direction cosines of a given vector.
- Explain what is meant by the vector projection of one vector onto another vector, and describe how to compute it.
- Calculate the work done by a given force.
If we apply a force to an object so that the object moves, we say that work is done by the force. Previously, we looked at a constant force and we assumed the force was applied in the direction of motion of the object. Under those conditions, work can be expressed as the product of the force acting on an object and the distance the object moves. In this chapter, however, we have seen that both force and the motion of an object can be represented by vectors.
In this section, we develop an operation called the dot product, which allows us to calculate work in the case when the force vector and the motion vector have different directions. The dot product essentially tells us how much of the force vector is applied in the direction of the motion vector. The dot product can also help us measure the angle formed by a pair of vectors and the position of a vector relative to the coordinate axes. It even provides a simple test to determine whether two vectors meet at a right angle.
The Dot Product and Its Properties
We have already learned how to add and subtract vectors. In this chapter, we investigate two types of vector multiplication. The first type of vector multiplication is called the dot product, based on the notation we use for it, and it is defined as follows:
Definition: dot product
The dot product of vectors \(\vecs{ u}=⟨u_1,u_2,u_3⟩\) and \(\vecs{ v}=⟨v_1,v_2,v_3⟩\) is given by the sum of the products of the components
\[\vecs{ u}⋅\vecs{ v}=u_1v_1+u_2v_2+u_3v_3. \nonumber \]
Note that if \(u\) and \(v\) are two-dimensional vectors, we calculate the dot product in a similar fashion. Thus, if \(\vecs{ u}=⟨u_1,u_2⟩\) and \(\vecs{ v}=⟨v_1,v_2⟩,\) then
\[\vecs{ u}⋅\vecs{ v}=u_1v_1+u_2v_2. \nonumber \]
When two vectors are combined under addition or subtraction, the result is a vector. When two vectors are combined using the dot product, the result is a scalar. For this reason, the dot product is often called the scalar product. It may also be called the inner product.
Example \(\PageIndex{1}\): Calculating Dot Products
- Find the dot product of \(\vecs{ u}=⟨3,5,2⟩\) and \(\vecs{ v}=⟨−1,3,0⟩\).
- Find the scalar product of \(\vecs{ p}=10\hat{\textbf i}−4 \hat{\textbf j}+7 \hat{\textbf k}\) and \(\vecs{ q}=−2\hat{\textbf i}+\hat{\textbf j}+6\hat{\textbf k}.\)
a. Substitute the vector components into the formula for the dot product:
\[ \begin{align*} \vecs{ u}⋅\vecs{ v} &=u_1v_1+u_2v_2+u_3v_3 \\[4pt] &=3(−1)+5(3)+2(0) \\[4pt] &=−3+15+0 \\[4pt] &=12. \end{align*}\]
b. The calculation is the same if the vectors are written using standard unit vectors. We still have three components for each vector to substitute into the formula for the dot product:
\[ \begin{align*} \vecs{ p}⋅\vecs{ q} &=p_1q_1+p_2q_2+p_3q_3 \\[4pt] &=10(−2)+(−4)(1)+(7)(6) \\[4pt] &=−20−4+42 \\[4pt] &=18.\end{align*}\]
Exercise \(\PageIndex{1}\)
Find \(\vecs{ u}⋅\vecs{ v}\), where \(\vecs{ u}=⟨2,9,−1⟩\) and \(\vecs{ v}=⟨−3,1,−4⟩.\)
- Hint
Multiply corresponding components and then add their products.
- Answer
Like vector addition and subtraction, the dot product has several algebraic properties. We prove three of these properties and leave the rest as exercises.
Properties of the Dot Product
Let \(\vecs{ u}\), \(\vecs{ v}\), and \(\vecs{ w}\) be vectors, and let \(c\) be a scalar.
- Commutative property \[\vecs{ u}⋅\vecs{ v}=\vecs{ v}⋅\vecs{ u} \nonumber \]
- Distributive property \[\vecs{ u}⋅(\vecs{ v}+\vecs{ w})=\vecs{ u}⋅\vecs{ v}+\vecs{ u}⋅\vecs{ w} \nonumber \]
- Associative property \[c(\vecs{ u}⋅\vecs{ v})=(c\vecs{ u})⋅\vecs{ v}=\vecs{ u}⋅(c\vecs{ v}) \nonumber \]
- Property of magnitude \[\vecs{ v}⋅\vecs{ v}=\|\vecs{ v}\|^2 \nonumber \]
Let \(\vecs{ u}=⟨u_1,u_2,u_3⟩\) and \(\vecs{ v}=⟨v_1,v_2,v_3⟩.\) Then
\[ \begin{align*} \vecs{ u}⋅\vecs{ v} &=⟨u_1,u_2,u_3⟩⋅⟨v_1,v_2,v_3⟩ \\[4pt] &=u_1v_1+u_2v_2+u_3v_3 \\[4pt] &= v_1u_1+v_2u_2+v_3u_3 \\[4pt] &= ⟨v_1,v_2,v_3⟩⋅⟨u_1,u_2,u_3⟩ \\[4pt] &=\vecs{ v}⋅\vecs{ u}.\end{align*}\]
The associative property looks like the associative property for real-number multiplication, but pay close attention to the difference between scalar and vector objects:
\[ \begin{align*} c(\vecs{ u}⋅\vecs{ v}) &=c(u_1v_1+u_2v_2+u_3v_3) \\[4pt] &=c(u_1v_1)+c(u_2v_2)+c(u_3v_3) \\[4pt] &=(cu_1)v_1+(cu_2)v_2+(cu_3)v_3 \\[4pt] &=⟨cu_1,cu_2,cu_3⟩⋅⟨v_1,v_2,v_3⟩ \\[4pt] &=c⟨u_1,u_2,u_3⟩⋅⟨v_1,v_2,v_3⟩ \\[4pt] &=(c\vecs{ u})⋅\vecs{ v}.\end{align*}\]
The proof that \(c(\vecs{ u}⋅\vecs{ v})=\vecs{ u}⋅(c\vecs{ v})\) is similar.
The fourth property shows the relationship between the magnitude of a vector and its dot product with itself:
\[ \begin{align*} \vecs{ v}⋅\vecs{ v} &=⟨v_1,v_2,v_3⟩⋅⟨v_1,v_2,v_3⟩\\[4pt] &=(v_1)^2+(v_2)^2+(v_3)^2 \\[4pt] &=\left[\sqrt{(v_1)^2+(v_2)^2+(v_3)^2}\right]^2 \\[4pt] &=\|\vecs{ v}\|^2.\end{align*}\]
Note that the definition of the dot product yields \(\vecs{ 0}⋅\vecs{ v}=0.\) By property iv. if \(\vecs{ v}⋅\vecs{ v}=0,\) then \(\vecs{ v}=\vecs{ 0}.\)
Example \(\PageIndex{2}\): Using Properties of the Dot Product
Let \(\vecs{ a}=⟨1,2,−3⟩\), \(\vecs{ b}=⟨0,2,4⟩\), and \( \vecs{ c} =⟨5,−1,3⟩\).
Find each of the following products.
- \(( \vecs{ a} ⋅ \vecs{ b}) \vecs{ c} \)
- \(\vecs{ a}⋅(2\vecs{ c})\)
- \(\|\vecs{ b}\|^2\)
a. Note that this expression asks for the scalar multiple of \(\vecs{ c}\) by \(\vecs{ a}⋅\vecs{ b}\):
\[ \begin{align*} (\vecs{ a}⋅\vecs{ b})\vecs{ c} &=(⟨1,2,−3⟩⋅⟨0,2,4⟩)⟨5,−1,3⟩ \\[4pt] &=(1(0)+2(2)+(−3)(4))⟨5,−1,3⟩ \\[4pt] &=−8⟨5,−1,3⟩ \\[4pt] &= ⟨−40,8,−24⟩.\end{align*}\]
b. This expression is a dot product of vector \(\vecs{ a}\) and scalar multiple 2\(\vecs{ c}\):
\[ \begin{align*} \vecs{ a}⋅(2\vecs{ c}) &=2(\vecs{ a}⋅\vecs{ c}) \\[4pt] &=2(⟨1,2,−3⟩⋅⟨5,−1,3⟩) \\[4pt] &=2(1(5)+2(−1)+(−3)(3)) \\[4pt] &=2(−6)=−12.\end{align*}\]
c. Simplifying this expression is a straightforward application of the dot product:
\[ \begin{align*} \|\vecs{ b}\|^2 &=\vecs{ b}⋅\vecs{ b} \\[4pt] &=⟨0,2,4⟩⋅⟨0,2,4⟩\\[4pt] &=0^2+2^2+4^2\\[4pt] &=0+4+16\\[4pt] &=20.\end{align*}\]
Exercise \(\PageIndex{2}\)
Find the following products for \(\vecs{ p}=⟨7,0,2⟩\), \(\vecs{ q}=⟨−2,2,−2⟩\), and \(\vecs{ r}=⟨0,2,−3⟩\).
- \((\vecs{ r}⋅\vecs{ p})\vecs{ q}\)
- \(\|\vecs{ p}\|^2\)
- Hint
\(\vecs{ r}⋅\vecs{ p}\) is a scalar.
- Answer
\(a. \quad (\vecs{ r}⋅\vecs{ p})\vecs{ q}=⟨12,−12,12⟩; \quad b. \quad \|\vecs{ p}\|^2=53\)
Using the Dot Product to Find the Angle between Two Vectors
When two nonzero vectors are placed in standard position, whether in two dimensions or three dimensions, they form an angle between them (Figure \(\PageIndex{1}\)). The dot product provides a way to find the measure of this angle. This property is a result of the fact that we can express the dot product in terms of the cosine of the angle formed by two vectors.
Evaluating a Dot Product
The dot product of two vectors is the product of the magnitude of each vector and the cosine of the angle between them:
\[\vecs{ u}⋅\vecs{ v}=‖\vecs{ u}‖‖\vecs{ v}‖\cos θ. \label{evaldot} \]
Place vectors \(\vecs{ u}\) and \(\vecs{ v}\) in standard position and consider the vector \(\vecs{ v}−\vecs{ u}\) (Figure \(\PageIndex{2}\)). These three vectors form a triangle with side lengths \(‖\vecs{ u}‖,‖\vecs{ v}‖\), and \(‖\vecs{ v}−\vecs{ u}‖\).
Recall from trigonometry that the law of cosines describes the relationship among the side lengths of the triangle and the angle \(θ\). Applying the law of cosines here gives
\[‖\vecs{ v}−\vecs{ u}‖^2=‖\vecs{ u}‖^2+‖\vecs{ v}‖^2−2‖\vecs{ u}‖‖\vecs{ v}‖\cos θ. \label{eq20} \]
The dot product provides a way to rewrite the left side of Equation \ref{eq20}:
\[ \begin{align*} ‖\vecs{ v}−\vecs{ u}‖^2 &=(\vecs{ v}−\vecs{ u})⋅(\vecs{ v}−\vecs{ u}) \\[4pt] &=(\vecs{ v}−\vecs{ u})⋅\vecs{ v}−(\vecs{ v}−\vecs{ u})⋅\vecs{ u} \\[4pt] &=\vecs{ v}⋅\vecs{ v}−\vecs{ u}⋅\vecs{ v}−\vecs{ v}⋅\vecs{ u}+\vecs{ u}⋅\vecs{ u} \\[4pt] &=\vecs{ v}⋅\vecs{ v}−\vecs{ u}⋅\vecs{ v}−\vecs{ u}⋅\vecs{ v}+\vecs{ u}⋅\vecs{ u} \\[4pt] &=‖\vecs{ v}‖^2−2\vecs{ u}⋅\vecs{ v}+‖\vecs{ u}‖^2.\end{align*}\]
Substituting into the law of cosines yields
\[ \begin{align*} ‖\vecs{ v}−\vecs{ u}‖^2 &=‖\vecs{ u}‖^2+‖\vecs{ v}‖^2−2‖\vecs{ u}‖‖\vecs{ v}‖\cos θ \\[4pt] ‖\vecs{ v}‖^2−2\vecs{ u}⋅\vecs{ v}+‖\vecs{ u}‖^2 &= ‖\vecs{ u}‖^2+‖\vecs{ v}‖^2−2‖\vecs{ u}‖‖\vecs{ v}‖\cos θ \\[4pt] −2\vecs{ u}⋅\vecs{ v} &=−2‖\vecs{ u}‖‖\vecs{ v}‖\cos θ \\[4pt] \vecs{ u}⋅\vecs{ v} &=‖\vecs{ u}‖‖\vecs{ v}‖\cos θ. \end{align*}\]
We can use the form of the dot product in Equation \ref{evaldot} to find the measure of the angle between two nonzero vectors by rearranging Equation \ref{evaldot} to solve for the cosine of the angle:
\[\cos θ=\dfrac{\vecs{ u}⋅\vecs{ v}}{‖\vecs{ u}‖‖\vecs{ v}‖}. \label{dot2} \]
Using this equation, we can find the cosine of the angle between two nonzero vectors. Since we are considering the smallest angle between the vectors, we assume \(0°≤θ≤180°\) (or \(0≤θ≤π\) if we are working in radians). The inverse cosine is unique over this range, so we are then able to determine the measure of the angle \(θ\).
Example \(\PageIndex{3}\): Finding the Angle between Two Vectors
Find the measure of the angle between each pair of vectors.
- \(\mathbf{\hat i} + \mathbf{\hat j} + \mathbf{\hat k}\) and \(2\mathbf{\hat i} – \mathbf{\hat j} – 3\mathbf{\hat k}\)
- \(⟨2,5,6⟩\) and \(⟨−2,−4,4⟩\)
a. To find the cosine of the angle formed by the two vectors, substitute the components of the vectors into Equation \ref{dot2}:
\[ \begin{align*} \cos θ &=\dfrac{(\mathbf{\hat i}+\mathbf{\hat j}+\mathbf{\hat k})⋅(2\mathbf{\hat i}−\mathbf{\hat j}−3\mathbf{\hat k})}{∥\mathbf{\hat i}+\mathbf{\hat j}+\mathbf{\hat k}∥⋅∥2\mathbf{\hat i}−\mathbf{\hat j}−3\mathbf{\hat k}∥} \\[4pt] &=\dfrac{1(2)+(1)(−1)+(1)(−3)}{\sqrt{1^2+1^2+1^2}\sqrt{2^2+(−1)^2+(−3)^2}} \\[4pt] &=\dfrac{−2}{\sqrt{3}\sqrt{14}} =\dfrac{−2}{\sqrt{42}}. \end{align*}\]
Therefore, \(θ=\arccos\dfrac{−2}{\sqrt{42}}\) rad.
b. Start by finding the value of the cosine of the angle between the vectors:
\[ \begin{align*} \cos θ &=\dfrac{⟨2,5,6⟩⋅⟨−2,−4,4⟩}{∥⟨2,5,6⟩∥⋅∥⟨−2,−4,4⟩∥} \\[4pt] &=\dfrac{2(−2)+(5)(−4)+(6)(4)}{\sqrt{2^2+5^2+6^2}\sqrt{(−2)^2+(−4)^2+4^2}} \\[4pt] &=\dfrac{0}{\sqrt{65}\sqrt{36}}=0.\end{align*}\]
Now, \(\cos θ=0\) and \(0≤θ≤π\), so \(θ=π/2\).
Exercise \(\PageIndex{3}\)
Find the measure of the angle, in radians, formed by vectors \(\vecs{ a}=⟨1,2,0⟩\) and \(\vecs{ b}=⟨2,4,1⟩\). Round to the nearest hundredth.
- Hint
Use the Equation \ref{dot2}.
- Answer
\(θ≈0.22\) rad
The angle between two vectors can be acute \((0<\cos θ<1),\) obtuse \((−1<\cos θ<0)\), or straight \((\cos θ=−1)\). If \(\cos θ=1\), then both vectors have the same direction. If \(\cos θ=0\), then the vectors, when placed in standard position, form a right angle (Figure \(\PageIndex{3}\)). We can formalize this result into a theorem regarding orthogonal (perpendicular) vectors.
Orthogonal Vectors
The nonzero vectors \(\vecs{u}\) and \(\vecs{v}\) are orthogonal vectors if and only if \(\vecs{u}⋅\vecs{v}=0.\)
Let \(\vecs{u}\) and \(\vecs{v}\) be nonzero vectors, and let \(θ\) denote the angle between them. First, assume \(\vecs{u}⋅\vecs{v}=0.\)Then
\[‖\vecs{u}‖‖\vecs{v}‖\cos θ=0. \nonumber \]
However, \(‖\vecs{u}‖≠0\) and \(‖\vecs{v}‖≠0,\) so we must have \(\cos θ=0\). Hence, \(θ=90°\), and the vectors are orthogonal.
Now assume \(\vecs{u}\) and \(\vecs{v}\) are orthogonal. Then \(θ=90°\) and we have
\[ \begin{align*} \vecs{u}⋅\vecs{v} &=‖\vecs{ u}‖‖\vecs{ v}‖\cos θ \\[4pt] &=‖\vecs{ u}‖‖\vecs{ v}‖\cos 90° \\[4pt] &=‖\vecs{ u}‖‖\vecs{ v}‖(0) \\[4pt] &=0. \end{align*}\]
The terms orthogonal, perpendicular, and normal each indicate that mathematical objects are intersecting at right angles. The use of each term is determined mainly by its context. We say that vectors are orthogonal and lines are perpendicular. The term normal is used most often when measuring the angle made with a plane or other surface.
Example \(\PageIndex{4}\): Identifying Orthogonal Vectors
Determine whether \(\vecs{p}=⟨1,0,5⟩\) and \(\vecs{q}=⟨10,3,−2⟩\) are orthogonal vectors.
Using the definition, we need only check the dot product of the vectors:
\[ \vecs{ p}⋅\vecs{ q}=1(10)+(0)(3)+(5)(−2)=10+0−10=0. \nonumber \]
Because \(\vecs{p}⋅\vecs{q}=0,\) the vectors are orthogonal (Figure \(\PageIndex{4}\)).
Exercise \(\PageIndex{4}\)
For which value of \(x\) is \(\vecs{ p}=⟨2,8,−1⟩\) orthogonal to \(\vecs{ q}=⟨x,−1,2⟩\)?
- Hint
Vectors \(\vecs{ p}\) and \(\vecs{ q}\) are orthogonal if and only if \(\vecs{ p}⋅\vecs{ q}=0\).
- Answer
Example \(\PageIndex{5}\): Measuring the Angle Formed by Two Vectors
Let \(\vecs{ v}=⟨2,3,3⟩.\) Find the measures of the angles formed by the following vectors.
- \(\vecs{ v}\) and \(\mathbf{\hat i}\)
- \(\vecs{ v}\) and \(\mathbf{\hat j}\)
- \(\vecs{ v}\) and \(\mathbf{\hat k}\)
a.Let α be the angle formed by \(\vecs{ v}\) and \(\mathbf{\hat i}\):
\[ \begin{align*} \cos{α} &=\dfrac{\vecs{ v}⋅\mathbf{\hat i}}{‖\vecs{ v}‖⋅\|\mathbf{\hat i}\|}=\dfrac{⟨2,3,3⟩⋅⟨1,0,0⟩}{\sqrt{2^2+3^2+3^2}\sqrt{1}}=\dfrac{2}{\sqrt{22}} \\[4pt] α &=\arccos\dfrac{2}{\sqrt{22}}≈1.130\,\text{rad.} \end{align*}\]
b. Let β represent the angle formed by \(\vecs{ v}\) and \(\mathbf{\hat j}\):
\[ \begin{align*} \cos{β} &=\dfrac{\vecs{ v}⋅\mathbf{\hat j}}{‖\vecs{ v}‖⋅\|\mathbf{\hat j}\|}=\dfrac{⟨2,3,3⟩⋅⟨0,1,0⟩}{\sqrt{2^2+3^2+3^2}\sqrt{1}}=\dfrac{3}{\sqrt{22}} \\[4pt] β &=\arccos\dfrac{3}{\sqrt{22}}≈0.877\,\text{rad.} \end{align*}\]
c. Let γ represent the angle formed by \(\vecs{ v}\) and \(\mathbf{\hat k}\):
\[ \begin{align*} \cos{γ} &=\dfrac{\vecs{ v}⋅\mathbf{\hat k}}{‖\vecs{ v}‖⋅\|\mathbf{\hat k}\|}=\dfrac{⟨2,3,3⟩⋅⟨0,0,1⟩}{\sqrt{2^2+3^2+3^2}\sqrt{1}}=\dfrac{3}{\sqrt{22}} \\[4pt]γ &=\arccos\dfrac{3}{\sqrt{22}}≈0.877\,\text{rad.} \end{align*}\]
Exercise \(\PageIndex{5}\)
Let \(\vecs{ v}=⟨3,−5,1⟩.\) Find the measure of the angles formed by each pair of vectors.
- \(\vecs{ v}\) and \(\mathbf{\hat i}\)
- \(\vecs{ v}\) and \(\mathbf{\hat j}\)
- \(\vecs{ v}\) and \(\mathbf{\hat k}\)
- Hint
\(\mathbf{\hat i}=⟨1,0,0⟩, \mathbf{\hat j}=⟨0,1,0⟩,\) and \(\mathbf{\hat k}=⟨0,0,1⟩\)
- Answer
\(a. α≈1.04\) rad; b. \(β≈2.58\) rad; c. \(γ≈1.40\) rad
The angle a vector makes with each of the coordinate axes, called a direction angle, is very important in practical computations, especially in a field such as engineering. For example, in astronautical engineering, the angle at which a rocket is launched must be determined very precisely. A very small error in the angle can lead to the rocket going hundreds of miles off course. Direction angles are often calculated by using the dot product and the cosines of the angles, called the direction cosines. Therefore, we define both these angles and their cosines.
Definition: direction angles
The angles formed by a nonzero vector and the coordinate axes are called the direction angles for the vector (Figure \(\PageIndex{5}\)). The cosines for these angles are called the direction cosines.
In Example \(\PageIndex{5}\), the direction cosines of \(\vecs{ v}=⟨2,3,3⟩\) are \(\cos α=\dfrac{2}{\sqrt{22}}, \cos β=\dfrac{3}{\sqrt{22}},\) and \(\cos γ=\dfrac{3}{\sqrt{22}}\). The direction angles of \(\vecs{ v}\) are \(α=1.130\) rad, \(β=0.877\) rad, and \(γ=0.877\) rad.
So far, we have focused mainly on vectors related to force, movement, and position in three-dimensional physical space. However, vectors are often used in more abstract ways. For example, suppose a fruit vendor sells apples, bananas, and oranges. On a given day, he sells 30 apples, 12 bananas, and 18 oranges. He might use a quantity vector, \(\vecs{ q}=⟨30,12,18⟩,\) to represent the quantity of fruit he sold that day. Similarly, he might want to use a price vector, \(\vecs{ p}=⟨0.50,0.25,1⟩,\) to indicate that he sells his apples for 50¢ each, bananas for 25¢ each, and oranges for $1 apiece. In this example, although we could still graph these vectors, we do not interpret them as literal representations of position in the physical world. We are simply using vectors to keep track of particular pieces of information about apples, bananas, and oranges.
This idea might seem a little strange, but if we simply regard vectors as a way to order and store data, we find they can be quite a powerful tool. Going back to the fruit vendor, let's think about the dot product, \(\vecs{ q}⋅\vecs{ p}\). We compute it by multiplying the number of apples sold (30) by the price per apple (50¢), the number of bananas sold by the price per banana, and the number of oranges sold by the price per orange. We then add all these values together. So, in this example, the dot product tells us how much money the fruit vendor had in sales on that particular day.
When we use vectors in this more general way, there is no reason to limit the number of components to three. What if the fruit vendor decides to start selling grapefruit? In that case, he would want to use four-dimensional quantity and price vectors to represent the number of apples, bananas, oranges, and grapefruit sold, and their unit prices. As you might expect, to calculate the dot product of four-dimensional vectors, we simply add the products of the components as before, but the sum has four terms instead of three.
Example \(\PageIndex{6}\): Using Vectors in an Economic Context
AAA Party Supply Store sells invitations, party favors, decorations, and food service items such as paper plates and napkins. When AAA buys its inventory, it pays 25¢ per package for invitations and party favors. Decorations cost AAA 50¢ each, and food service items cost 20¢ per package. AAA sells invitations for $2.50 per package and party favors for $1.50 per package. Decorations sell for $4.50 each and food service items for $1.25 per package.
During the month of May, AAA Party Supply Store sells 1258 invitations, 342 party favors, 2426 decorations, and 1354 food service items. Use vectors and dot products to calculate how much money AAA made in sales during the month of May. How much did the store make in profit?
The cost, price, and quantity vectors are
\[ \begin{align*} \vecs{ c} &=⟨0.25,0.25,0.50,0.20⟩ \\[4pt] \vecs{ p} &=⟨2.50,1.50,4.50,1.25⟩ \\[4pt] \vecs{ q} &=⟨1258,342,2426,1354⟩. \end{align*}\]
AAA sales for the month of May can be calculated using the dot product \(\vecs{ p}⋅\vecs{ q}\). We have
\[ \begin{align*} \vecs{ p}⋅\vecs{ q} &=⟨2.50,1.50,4.50,1.25⟩⋅⟨1258,342,2426,1354⟩ \\[4pt] &=3145+513+10917+1692.5 \\[4pt] &= 16267.5. \end{align*}\]
So, AAA took in $16,267.50 during the month of May. To calculate the profit, we must first calculate how much AAA paid for the items sold. We use the dot product \(\vecs{c}⋅\vecs{q}\) to get
\[ \begin{align*} \vecs{ c}⋅\vecs{ q} &=⟨0.25,0.25,0.50,0.20⟩⋅⟨1258,342,2426,1354⟩ \\[4pt] &=314.5+85.5+1213+270.8 \\[4pt] &=1883.8. \end{align*}\]
So, AAA paid $1,883.80 for the items they sold. Their profit, then, is given by
\[\vecs{ p}⋅\vecs{ q}−\vecs{ c}⋅\vecs{ q}=16267.5−1883.8 =14383.7. \nonumber \]
Therefore, AAA Party Supply Store made $14,383.70 in May.
Exercise \(\PageIndex{6}\)
On June 1, AAA Party Supply Store decided to increase the price they charge for party favors to $2 per package. They also changed suppliers for their invitations, and are now able to purchase invitations for only 10¢ per package. All their other costs and prices remain the same. If AAA sells 1408 invitations, 147 party favors, 2112 decorations, and 1894 food service items in the month of June, use vectors and dot products to calculate their total sales and profit for June.
- Hint
Use four-dimensional vectors for cost, price, and quantity sold.
- Answer
Sales = $15,685.50; profit = $14,073.15
As we have seen, addition combines two vectors to create a resultant vector. But what if we are given a vector and we need to find its component parts? We use vector projections to perform the opposite process; they can break down a vector into its components. The magnitude of a vector projection is a scalar projection. For example, if a child is pulling the handle of a wagon at a 55° angle, we can use projections to determine how much of the force on the handle is actually moving the wagon forward (\(\PageIndex{6}\)). We return to this example and learn how to solve it after we see how to calculate projections.
Definition: Vector and Projection
The vector projection of \(\vecs{ v}\) onto \(\vecs{ u}\) is the vector labeled \(\text{proj}_\vecs{ u}\vecs{ v}\) in Figure \(\PageIndex{7}\). It has the same initial point as \(\vecs{ u}\) and \(\vecs{ v}\) and the same direction as \(\vecs{ u}\), and represents the component of \(\vecs{ v}\) that acts in the direction of \(\vecs{ u}\). If \(θ\) represents the angle between \(\vecs{ u}\) and \(\vecs{ v}\), then, by properties of triangles, we know the length of \(\text{proj}_\vecs{ u}\vecs{ v}\) is \(\|\text{proj}_\vecs{ u}\vecs{ v}\|=‖\vecs{ v}‖\cos θ.\) When expressing \(\cos θ\) in terms of the dot product, this becomes
\[ \|\text{proj}_\vecs{ u}\vecs{ v}\|=‖\vecs v‖\cos θ=‖\vecs{ v}‖\left(\dfrac{\vecs{ u}⋅\vecs{ v}}{‖\vecs{ u}‖‖\vecs{ v}‖}\right)=\dfrac{\vecs{ u}⋅\vecs{ v}}{‖\vecs{ u}‖.} \nonumber \]
We now multiply by a unit vector in the direction of \(\vecs{ u}\) to get \(\text{proj}_\vecs{ u}\vecs{ v}\):
\[\text{proj}_\vecs{ u}\vecs{ v}=\dfrac{\vecs{ u}⋅\vecs{ v}}{‖\vecs{ u}‖}\left(\dfrac{1}{‖\vecs{ u}‖}\vecs{ u}\right)=\dfrac{\vecs{ u}⋅\vecs{ v}}{‖\vecs{ u}‖^2}\vecs{ u}. \nonumber \]
The length of this vector is also known as the scalar projection of \(\vecs{ v}\) onto \(\vecs{ u}\) and is denoted by
\[\|\text{proj}_\vecs{ u}\vecs{ v}\|=\text{comp}_\vecs{ u}\vecs{ v}=\dfrac{\vecs{ u}⋅\vecs{ v}}{‖\vecs{ u}‖.} \nonumber \]
Example \(\PageIndex{7}\): Finding Projections
Find the projection of \(\vecs{ v}\) onto \(\vecs{ u}\).
- \(\vecs{v}=⟨3,5,1⟩\) and \(\vecs{u}=⟨−1,4,3⟩\)
- \(\vecs{v}=3\mathbf{\hat i}−2\mathbf{\hat j}\) and \(\vecs{u}=\mathbf{\hat i}+6\mathbf{\hat j}\)
a. Substitute the components of \(\vecs{ v}\) and \(\vecs{ u}\) into the formula for the projection:
\[\begin{align*} \text{proj}_\vecs{ u}\vecs{ v} &=\dfrac{\vecs{ u}⋅\vecs{ v}}{‖\vecs{ u}‖^2}\vecs{ u} \\[4pt] &=\dfrac{⟨−1,4,3⟩⋅⟨3,5,1⟩}{∥⟨−1,4,3⟩∥^2}⟨−1,4,3⟩ \\[4pt] &=\dfrac{−3+20+3}{(−1)^2+4^2+3^2}⟨−1,4,3⟩ \\[4pt] &=\dfrac{20}{26}⟨−1,4,3⟩ \\[4pt] &=⟨−\dfrac{10}{13},\dfrac{40}{13},\dfrac{30}{13}⟩. \end{align*}\]
b. To find the two-dimensional projection, simply adapt the formula to the two-dimensional case:
\[\begin{align*} \text{proj}_\vecs{ u}\vecs{ v} &=\dfrac{\vecs{ u}⋅\vecs{ v}}{‖\vecs{ u}‖^2}\vecs{ u} \\[4pt] &=\dfrac{(\mathbf{\hat i}+6\mathbf{\hat j})⋅(3\mathbf{\hat i}−2\mathbf{\hat j})}{∥\mathbf{\hat i}+6\mathbf{\hat j}∥^2}(\mathbf{\hat i}+6\mathbf{\hat j}) \\[4pt] &= \dfrac{1(3)+6(−2)}{1^2+6^2}(\mathbf{\hat i}+6\mathbf{\hat j}) \\[4pt] &= −\dfrac{9}{37}(\mathbf{\hat i}+6\mathbf{\hat j}) \\[4pt] &= −\dfrac{9}{37}\mathbf{\hat i}−\dfrac{54}{37}\mathbf{\hat j}.\end{align*}\]
Sometimes it is useful to decompose vectors—that is, to break a vector apart into a sum. This process is called the resolution of a vector into components. Projections allow us to identify two orthogonal vectors having a desired sum. For example, let \(\vecs{ v}=⟨6,−4⟩\) and let \(\vecs{ u}=⟨3,1⟩.\) We want to decompose the vector \(\vecs{ v}\) into orthogonal components such that one of the component vectors has the same direction as \(\vecs{ u}\).
We first find the component that has the same direction as \(\vecs{ u}\) by projecting \(\vecs{ v}\) onto \(\vecs{ u}\). Let \(\vecs{ p}=\text{proj}_\vecs{ u}\vecs{ v}\). Then, we have
\[\begin{align*}\vecs{ p} =\dfrac{\vecs{ u}⋅\vecs{ v}}{‖\vecs{ u}‖^2}\vecs{ u} \\[4pt] = \dfrac{18−4}{9+1}\vecs{ u} \\[4pt] = \dfrac{7}{5}\vecs{ u}=\dfrac{7}{5}⟨3,1⟩=⟨\dfrac{21}{5},\dfrac{7}{5}⟩. \end{align*}\]
Now consider the vector \(\vecs{ q}=\vecs{ v}−\vecs{ p}.\) We have
\[\begin{align*} \vecs{ q} =\vecs{ v}−\vecs{ p} \\[4pt] = ⟨6,−4⟩−⟨\dfrac{21}{5},\dfrac{7}{5}⟩ \\[4pt] = ⟨\dfrac{9}{5},−\dfrac{27}{5}⟩. \end{align*}\]
Clearly, by the way we defined \(\vecs{ q}\), we have \(\vecs{ v}=\vecs{ q}+\vecs{ p},\) and
\[\begin{align*}\vecs{ q}⋅\vecs{ p} =⟨\dfrac{9}{5},−\dfrac{27}{5}⟩⋅⟨\dfrac{21}{5},\dfrac{7}{5}⟩ \\[4pt] = \dfrac{9(21)}{25}+−\dfrac{27(7)}{25} \\[4pt] = \dfrac{189}{25}−\dfrac{189}{25}=0. \end{align*}\]
Therefore, \(\vecs{ q}\) and \(\vecs{ p}\) are orthogonal.
Example \(\PageIndex{8}\): Resolving Vectors into Components
Express \(\vecs{ v}=⟨8,−3,−3⟩\) as a sum of orthogonal vectors such that one of the vectors has the same direction as \(\vecs{ u}=⟨2,3,2⟩.\)
Let \(\vecs{ p}\) represent the projection of \(\vecs{ v}\) onto \(\vecs{ u}\):
\[ \begin{align*} \vecs{ p} &=\text{proj}_\vecs{ u}\vecs{ v} \\[4pt] &=\dfrac{\vecs{ u}⋅\vecs{ v}}{‖\vecs{ u}‖^2}\vecs{ u} \\[4pt] &=\dfrac{⟨2,3,2⟩⋅⟨8,−3,−3⟩}{∥⟨2,3,2⟩∥^2}⟨2,3,2⟩ \\[4pt] &=\dfrac{16−9−6}{2^2+3^2+2^2}⟨2,3,2⟩ \\[4pt] &=\dfrac{1}{17}⟨2,3,2⟩ \\[4pt] &=⟨\dfrac{2}{17},\dfrac{3}{17},\dfrac{2}{17}⟩. \end{align*} \nonumber \]
\[ \begin{align*} \vecs{ q} &=\vecs{ v}−\vecs{ p}=⟨8,−3,−3⟩−⟨\dfrac{2}{17},\dfrac{3}{17},\dfrac{2}{17}⟩\\[4pt] &=⟨\dfrac{134}{17},−\dfrac{54}{17},−\dfrac{53}{17}⟩. \end{align*} \nonumber \]
To check our work, we can use the dot product to verify that \(\vecs{ p}\) and \(\vecs{ q}\) are orthogonal vectors:
\[ \begin{align*}\vecs{ p}⋅\vecs{ q}&=⟨\dfrac{2}{17},\dfrac{3}{17},\dfrac{2}{17}⟩⋅⟨\dfrac{134}{17},−\dfrac{54}{17},−\dfrac{53}{17}⟩\\[4pt] &=\dfrac{268}{289}−\dfrac{162}{289}−\dfrac{106}{289}=0. \end{align*} \nonumber \]
\[\vecs{ v}=\vecs{ p}+\vecs{ q}=⟨\dfrac{2}{17},\dfrac{3}{17},\dfrac{2}{17}⟩+⟨\dfrac{134}{17},−\dfrac{54}{17},−\dfrac{53}{17}⟩. \nonumber \]
Exercise \(\PageIndex{7}\)
Express \(\vecs{ v}=5\mathbf{\hat i}−\mathbf{\hat j}\) as a sum of orthogonal vectors such that one of the vectors has the same direction as \(\vecs{ u}=4\mathbf{\hat i}+2\mathbf{\hat j}\).
- Hint
Start by finding the projection of \(\vecs{ v}\) onto \(\vecs{ u}\).
- Answer
\(\vecs{ v}=\vecs{ p}+\vecs{ q},\) where \(\vecs{ p}=\dfrac{18}{5}\mathbf{\hat i}+\dfrac{9}{5}\mathbf{\hat j}\) and \(\vecs{ q}=\dfrac{7}{5}\mathbf{\hat i}−\dfrac{14}{5}\mathbf{\hat j}\)
Example \(\PageIndex{9}\): Scalar Projection of Velocity
A container ship leaves port traveling \(15°\) north of east. Its engine generates a speed of 20 knots along that path (see the following figure). In addition, the ocean current moves the ship northeast at a speed of 2 knots. Considering both the engine and the current, how fast is the ship moving in the direction \(15°\) north of east? Round the answer to two decimal places.
Let \(\vecs{ v}\) be the velocity vector generated by the engine, and let \(\vecs{w}\) be the velocity vector of the current. We already know \(‖\vecs{ v}‖=20\) along the desired route. We just need to add in the scalar projection of \(\vecs{ w}\) onto \(\vecs{ v}\). We get
\[ \begin{align*} \text{comp}_\vecs{ v}\vecs{ w}=\dfrac{\vecs{ v}⋅\vecs{ w}}{‖\vecs{ v}‖} \\[4pt] =\dfrac{‖\vecs{ v}‖‖\vecs{ w}‖\cos(30°)}{‖\vecs{ v}‖} =‖\vecs{ w}‖\cos(30°) =2\dfrac{\sqrt{3}}{2}=\sqrt{3}≈1.73\,\text{knots.}\end{align*}\]
The ship is moving at 21.73 knots in the direction \(15°\) north of east.
Exercise \(\PageIndex{8}\)
Repeat the previous example, but assume the ocean current is moving southeast instead of northeast, as shown in the following figure.
- Hint
Compute the scalar projection of \(\vecs{ w}\) onto \(\vecs{ v}\).
- Answer
21 knots
Now that we understand dot products, we can see how to apply them to real-life situations. The most common application of the dot product of two vectors is in the calculation of work.
From physics, we know that work is done when an object is moved by a force. When the force is constant and applied in the same direction the object moves, then we define the work done as the product of the force and the distance the object travels: \(W=Fd\). We saw several examples of this type in earlier chapters. Now imagine the direction of the force is different from the direction of motion, as with the example of a child pulling a wagon. To find the work done, we need to multiply the component of the force that acts in the direction of the motion by the magnitude of the displacement. The dot product allows us to do just that. If we represent an applied force by a vector \(\vecs{ F}\) and the displacement of an object by a vector \(\vecs{ s}\), then the work done by the force is the dot product of \(\vecs{ F}\) and \(\vecs{ s}\).
Definition: Constant Force
When a constant force is applied to an object so the object moves in a straight line from point \(P\) to point \(Q\), the work \(W\) done by the force \(\vecs{ F}\), acting at an angle θ from the line of motion, is given by
\[W=\vecs{ F}⋅\vecd{PQ}=∥\vecs{ F}∥∥\vecd{PQ}∥\cos θ. \nonumber \]
Let's revisit the problem of the child's wagon introduced earlier. Suppose a child is pulling a wagon with a force having a magnitude of 8 lb on the handle at an angle of 55°. If the child pulls the wagon 50 ft, find the work done by the force (Figure \(\PageIndex{8}\)).
We have
\[W=∥\vecs{ F}∥∥\vecd{PQ}∥\cos θ=8(50)(\cos(55°))≈229\,\text{ft⋅lb.} \nonumber \]
In U.S. standard units, we measure the magnitude of force \(∥\vecs{ F}∥\) in pounds. The magnitude of the displacement vector \(∥\vecd{PQ}∥\) tells us how far the object moved, and it is measured in feet. The customary unit of measure for work, then, is the foot-pound. One foot-pound is the amount of work required to move an object weighing 1 lb a distance of 1 ft straight up. In the metric system, the unit of measure for force is the newton (N), and the unit of measure of magnitude for work is a newton-meter (N·m), or a joule (J).
Example \(\PageIndex{10}\): Calculating Work
A conveyor belt generates a force \(\vecs{ F}=5\mathbf{\hat i}−3\mathbf{\hat j}+\mathbf{\hat k}\) that moves a suitcase from point \((1,1,1)\) to point \((9,4,7)\) along a straight line. Find the work done by the conveyor belt. The distance is measured in meters and the force is measured in newtons.
The displacement vector \(\vecd{PQ}\) has initial point \((1,1,1)\) and terminal point \((9,4,7)\):
\[\vecd{PQ}=⟨9−1,4−1,7−1⟩=⟨8,3,6⟩=8\mathbf{\hat i}+3\mathbf{\hat j}+6\mathbf{\hat k}. \nonumber \]
Work is the dot product of force and displacement:
\[\begin{align*} W &=\vecs{ F}⋅\vecd{PQ} \\[4pt] &= (5\mathbf{\hat i}−3\mathbf{\hat j}+\mathbf{\hat k})⋅(8\mathbf{\hat i}+3\mathbf{\hat j}+6\mathbf{\hat k}) \\[4pt] = 5(8)+(−3)(3)+1(6) \\[4pt] &=37\,\text{N⋅m} \\[4pt] &= 37\,\text{J} \end{align*}\]
Exercise \(\PageIndex{9}\)
A constant force of 30 lb is applied at an angle of 60° to pull a handcart 10 ft across the ground. What is the work done by this force?
- Hint
Use the definition of work as the dot product of force and distance.
- Answer
150 ft-lb
Key Concepts
- The dot product, or scalar product, of two vectors \(\vecs{ u}=⟨u_1,u_2,u_3⟩\) and \(\vecs{ v}=⟨v_1,v_2,v_3⟩\) is \(\vecs{ u}⋅\vecs{ v}=u_1v_1+u_2v_2+u_3v_3\).
- The dot product satisfies the following properties:
- \(\vecs{ u}⋅\vecs{ v}=\vecs{ v}⋅\vecs{ u}\)
- \(\vecs{ u}⋅(\vecs{ v}+\vecs{ w})=\vecs{ u}⋅\vecs{ v}+\vecs{ u}⋅\vecs{ w}\)
- \(c(\vecs{ u}⋅\vecs{ v})=(c\vecs{ u})⋅\vecs{ v}=\vecs{ u}⋅(c\vecs{ v})\)
- \(\vecs{ v}⋅\vecs{ v}=‖\vecs{ v}‖^2\)
- The dot product of two vectors can be expressed, alternatively, as \(\vecs{ u}⋅\vecs{ v}=‖\vecs{ u}‖‖\vecs{ v}‖\cos θ.\) This form of the dot product is useful for finding the measure of the angle formed by two vectors.
- Vectors \(\vecs{ u}\) and \(\vecs{ v}\) are orthogonal if \(\vecs{ u}⋅\vecs{ v}=0\).
- The angles formed by a nonzero vector and the coordinate axes are called the direction angles for the vector. The cosines of these angles are known as the direction cosines.
- The vector projection of \(\vecs{ v}\) onto \(\vecs{ u}\) is the vector \(\text{proj}_\vecs{ u}\vecs{ v}=\dfrac{\vecs{ u}⋅\vecs{ v}}{‖\vecs{ u}‖^2}\vecs{ u}\). The magnitude of this vector is known as the scalar projection of \(\vecs{ v}\) onto \(\vecs{ u}\), given by \(\text{comp}_\vecs{ u}\vecs{ v}=\dfrac{\vecs{ u}⋅\vecs{ v}}{‖\vecs{ u}‖}\).
- Work is done when a force is applied to an object, causing displacement. When the force is represented by the vector \(\vecs{ F}\) and the displacement is represented by the vector \(\vecs{ s}\), then the work done \(W\) is given by the formula \(W=\vecs{ F}⋅\vecs{ s}=∥\vecs{ F}∥‖\vecs{ s}‖\cos θ.\)
Key Equations
- Dot product of \(\vecs{ u}\) and \(\vecs{ v}\)
\(\vecs{ u}⋅\vecs{ v}=u_1v_1+u_2v_2+u_3v_3=‖\vecs{ u}‖‖\vecs{ v}‖\cos θ\)
- Cosine of the angle formed by \(\vecs{ u}\) and \(\vecs{ v}\)
\(\cos θ=\dfrac{\vecs{ u}⋅\vecs{ v}}{‖\vecs{ u}‖‖\vecs{ v}‖}\)
- Vector projection of \(\vecs{ v}\) onto \(\vecs{ u}\)
\(\text{proj}_\vecs{ u}\vecs{ v}=\dfrac{\vecs{ u}⋅\vecs{ v}}{‖\vecs{ u}‖^2}\vecs{ u}\)
- Scalar projection of \(\vecs{ v}\) onto \(\vecs{ u}\)
\(\text{comp}_\vecs{ u}\vecs{ v}=\dfrac{\vecs{ u}⋅\vecs{ v}}{‖\vecs{ u}‖}\)
- Work done by a force \(\vecs{ F}\) to move an object through displacement vector \(\vecd{PQ}\)
\(W=\vecs{ F}⋅\vecd{PQ}=∥\vecs{ F}∥∥\vecd{PQ}∥\cos θ\)
- direction angles
- the angles formed by a nonzero vector and the coordinate axes
- direction cosines
- the cosines of the angles formed by a nonzero vector and the coordinate axes
- dot product or scalar product
- \(\vecs{ u}⋅\vecs{ v}=u_1v_1+u_2v_2+u_3v_3\) where \(\vecs{ u}=⟨u_1,u_2,u_3⟩\) and \(\vecs{ v}=⟨v_1,v_2,v_3⟩\)
- scalar projection
- the magnitude of the vector projection of a vector
- orthogonal vectors
- vectors that form a right angle when placed in standard position
- vector projection
- the component of a vector that follows a given direction
- work done by a force
- work is generally thought of as the amount of energy it takes to move an object; if we represent an applied force by a vector \(\vecs{ F}\) and the displacement of an object by a vector \(\vecs{ s}\), then the work done by the force is the dot product of \(\vecs{ F}\) and \(\vecs{ s}\).
Contributors and Attributions
Gilbert Strang (MIT) and Edwin "Jed" Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.
- edited for vector notation by Paul Seeburger


text:
Customers searching to book a photograph booth frequently ask what's best: digital or old-fashioned film. The reply is really to the person. Each kind has its own pros and cons.
The classic, vintage type of a movie booth provides them a higher "awesome" factor, as well as their photos look great. Regrettably, because of the rise of digital photo booths, film booths have become progressively difficult to locate. Should you choose find a movie photo booth for the event, there are several additional factors you should know of prior to you making your choice.
Film booths are slower to print than digital booths it requires them about 3 minutes to build up a photograph. They're also more costly to function and, evaluating to 900 lbs, more costly to move and hard to create inside some structures. These added costs make film booths more costly to book than the usual digital booth.
Whenever you have a picture inside a film photo booth, the resulting photo strip is a-of-a-kind. Film booths are not able to keep a duplicate from the images. Because the operation is completely mechanical, there aren't any stored files with no negatives. Each strip that's developed is exclusive there should never be another copy. This belongs to why is the show booth experience special, however it might have its drawbacks if you're trying to construct a scrapbook of the event. If you are using a movie booth as well as your visitors don't make you a duplicate for scrapbook, then you'll never see their pictures.
Nowadays digital photo booth rules. The low cost to function makes rentals much more affordable. Visitors just wait about ten seconds to get their photo strip, and you may even obtain a CD in the finish from the night with all the crazy pictures both you and your buddies required. Make certain to book a photograph booth that prints two strips of photos so that your visitors can share or make you a duplicate.
Digital photo booths are occasionally considered to have poor picture quality, however this status is undeserved. It is a fact that the majority of the booths that you simply find at malls and theme parks nowadays have cheap, tacky designs and grainy photo quality. Some manufacturers appear to possess lost touch using the essence from the photo booth.
The good thing is that the digital photo booth does not need to compromise quality or style. There's a couple of companies emerging round the country which have honored the nostalgic excellence of the classic photo booth, but combined by using the characteristics that individuals love inside a digital photobooth. These digital booths always produce great pictures, and could look much more classic than a few of the "classic" film photo booths.

subset: commoncrawl


text:
"Copyright for Librarians" (CFL) is an online open curriculum on copyright law that was developed jointly with Harvard's Berkman Center for Internet and Society.
The goal is to provide librarians in developing and transition countries with information concerning copyright law. More specifically, it aspires to inform librarians concerning copyright law in general; the aspects of copyright law that most affect libraries; and how librarians in the future could most effectively participate in the processes by which copyright law is interpreted and shaped.
The curriculum was translated into Arabic by a team at the Egyptian Universities Libraries Consortium (EULC).


text:
State Sen. John Sampson charged with embezzlement
State Sen. John Sampson has been indicted on charges of embezzlement, obstruction of justice and lying to the FBI.
Sampson was a court-appointed referee to watch over escrow accounts for sales of foreclosed properties in Brooklyn. It's alleged he embezzled $440,000 between 1998 and 2008.
He's also accused of funneling funds into his failed campaign for Brooklyn DA.
There was no immediate response to a message left Monday for Sampson's attorney.
Senate Democratic Conference Leader Andrea Stewart-Cousins said Sampson has been stripped of his ranking positions and committee assignments.
The arrest came less than a week after prosecutors revealed that former Sen. Shirley Huntley made numerous secret recordings of other elected officials in a bid for leniency in her own corruption case.


text:
Coil Basket
Coiling technique was introduced in the 1920s at Goulburn Island to the Maung people by missionaries and quickly spread to the mainland. Many artists produce coiled baskets of varied shapes, ranging from small round baskets to large oval baby baskets made from dyed pandanus. Artists combine colours and patterns to obtain intricate new graphic patterns.


text:
Blockchain technology is transforming the real estate market – Cointelegraph Magazine
Property is the world's single largest store of wealth, and if the cryptocurrency and blockchain world is seeking an express route to mass adoption, it could do worse than partnering with the real estate industry.
According to a September 2021 report by Savills World Research, the estimated value of all the world's real estate stands at $326.5 trillion. By comparison, crypto-sector market capitalization was about $1 trillion in mid-July.
The property market, moreover — at least its commercial real estate segment — is also characterized by costly entry barriers and asymmetrical information that favor insiders. Its fees are high, paperwork onerous, and deeds are sometimes defective, falsified or missing. Some properties can take years to move — another way of saying its market is illiquid. All in all, it isn't surprising that many believe this market is ripe for disruption, particularly through blockchain-enabled tokenization.
This notion of tokenizing real estate isn't entirely new. As far back as 2019, for example, a 6.5-million-euro villa in Boulogne, outside Paris, was tokenized. One million shares were put up for sale on the Ethereum blockchain, the first property in France ever sold as a blockchain transaction. An individual could have purchased a part of the luxury villa for as little as 6.5 euros.
Will everything be fractionalized?
Last year's nonfungible token (NFT) breakout — and real properties are nonfungible, i.e., not interchangeable — along with some more supportive regulations, like Regulation Crowdfunding (Reg CF) in the United States, have trained the spotlight more squarely on crypto and property partnerships. This year's metaverse hype, including Yugo Labs' record-breaking virtual land auction, has not discouraged activity in the real property world, either.
"Web3 will be all about ownership, owning fractionalized shares," says Bobby Singh, founder of the NiftySky DAO, speaking at June's NFT.NYC 2022 convention, which featured an entire track on tokenized real estate. "Imagine fractionalizing the Empire State Building into 2 billion shares." An individual could own a piece of the Empire State Building for several dollars.
Ownership creates its own momentum, Singh continued. "If you become a collector, an owner, you're more likely to talk about it." More owners mean more excitement. "The concept of title is very important."
"Blockchain has the potential to transform real estate," Lamont Black, associate professor in DePaul University's department of finance and real estate, tells Magazine. Real estate is all about records of ownership and how a property is financed, he explains, and "blockchain is ideally suited as a shared system of record-keeping for this type of application."
Many of these principles are "already being applied to digital real estate in the metaverse," adds Black, while the ideas behind Web3 — of which the metaverse is one part — "are very much rooted in ownership of digital assets, including basic things like personal data."
"The efficiency and certainty that comes with tokenization is undeniable," David Tawil, president and co-founder at ProChain Capital, tells Magazine, and this hasn't been lost on the real estate industry.
A market that dwarfs the cryptoverse
If one accepts Savills' numbers, the value of the world's real estate is more than 300 times the size of the crypto and blockchain sector, which recently slumped below $1 trillion in market capitalization. That disparity hasn't been lost on observers.
"If even just 0.5% of the total $280 trillion global property market were tokenised in the next five years, it would become a $1.4 trillion market," wrote Moore Global, a global accountancy, advisory and consulting network in August.
Alternatively, if one uses Savills' estimate of a $327-trillion market: If just 1% of the global real estate market were tokenized, it would triple the current market cap of the entire cryptocurrency world.
About four-fifths of the world's real estate is residential, according to Savills. Commercial real estate accounts for only about one-tenth of the total, but that might be where tokenization first makes an impact, some say.
Singh, a veteran of the New York commercial real estate business, explained at the NFT.NYC convention that the legacy commercial real estate market has "a lot of friction" and is burdened by a lack of liquidity. Innovations like blockchain-based NFTs can help with record-keeping because the blockchain is transparent, "and fractionalization will make real estate more liquid."
"We believe this market will be more open to alternate sources of capital raising, including tokenization," Navonil Roy, CEO of United Arab Emirates-based LandOrc, tells Magazine. His firm facilitates lending for the real estate industry by providing access to decentralized financing, using land titles in nonfungible token form as a collateral.
Capital formation is often an obstacle in real estate ventures, and tokenization can "open the door for a broader pool of investors," Sean Stein Smith, assistant professor in the department of economics and business at Lehman College, tells Magazine, "by being able to tokenize and bifurcate the ownership and custody of the underlying physical real estate asset." It can also enable peer-to-peer secondary transactions so "a robust second market can also be developed."
The fact that crypto transactions are conducted in real-time offers potential advantages, too, such as "increasing the speed with which mortgages are approved and transactions are completed," adds Stein Smith.
Obstacles remain
Despite the enormous potential, blockchain technology has made relatively small inroads in the area of property rights so far. The Boulogne villa cited above was more of an exception than a rule.
"Blockchain is a technology that requires coordination among market participants. Until there is more adoption of this technology, the impact will be limited," Black tells Magazine, adding:
"Another hurdle is the role of government. Because real estate title is largely regulated by local governments, the recording of ownership on a blockchain will require government adoption as well. The forward-thinking and nimble municipalities will lead the way."
The "promise" of a globalized, tokenized real estate market with secondary market trading has taken some time "because it required getting multiple licenses," Max Dilendorf, partner at the Dilendorf Law Firm and who has been working on real estate tokenization projects since 2017, tells Magazine. Securitize LLC led the way, he says, becoming a U.S. Securities and Exchange Commission-registered transfer agent operating on the blockchain about three years ago, but "companies have spent years and years to get licenses in the U.S." as well as Asia and Europe.
Another obstacle is that so much of the required data to complete a tokenized real estate transaction does not occur natively on the blockchain. It has to be entered manually. University of Basel professor Fabian Schär, for example, wanted to invest a few hundred Swiss francs in a multifamily house in Detroit, as he recounted in a May 2022 Credit Suisse Insights interview:
"'The technical process related to the token functioned seamlessly.' But then came something that made Schär cancel the transaction: 'There were around 150 pages of legal documents that I had to read and sign.'"
The real estate sector, too, is sometimes resistant to change, which could impede adoption. "Real estate brokers and agents are relatively conservative in their adoption to new technology due to both the monetary sums involved and the implications associated with property ownership rights," notes Stein Smith.
More success stories may be required, too, before things really get moving. "Real estate owners looking to raise money have a single goal: lower cost of capital," Yael Tamar, CEO and co-founder of SolidBlock — a real estate tokenization platform — tells Magazine. "Until they are convinced that there is an audience of investors looking specifically for property-backed security tokens, they will not bother with nice-to-have solutions."
Then, too, the DeFi summer may have inadvertently slowed things down, with DeFi lenders becoming "spoiled" by the unusually high lending returns they enjoyed during this highly liquid period, suggests Roy, adding:
"Their expectations of returns in the digital asset world cannot be replicated with real world assets, which are grounded by real world economics. This change in mindset is the most daunting obstacle."
The path ahead could be long, too. When Savills' Marie Hickey, director of U.K. commercial research, recently wrote about "four trends shaping retail real estate globally," there was no mention at all of tokenization or blockchain technology. Asked about this, Hickey tells Magazine, "Tokenized real estate is just too insignificant to be cited as a key trend at this point."
Positive regulatory developments in the U.S.
Still, recent regulation changes in the U.S. could prompt a boom in tokenized projects in Dilendorf's view. Reg CF, whose fundraising cap was boosted from $1 million to $5 million in late 2020, "will pave the way," predicts Dilendorf. Reg CF enables a company to fundraise among both accredited and non-accredited investors, and there is no cap on the number of investors who can participate.
Meanwhile, alternative trading systems (ATS) based in the U.S. are building bridges to ATS platforms in Europe and Asia. "So, now if you put your digital real estate token on a platform in the U.S., an investor from Europe or Asia can participate in a secondary market trade," Dilendorf tells Magazine. This could boost the secondary market substantially.
Transaction fees are low on these blockchain-based ATS platforms like Securitize as well, while smart contracts ensure that transfers are executed between Know Your Customer-verified accounts only. "It's the creation of a new way to raise capital for small firms," which is almost like an initial public offering, adds Dilendorf.
NFTs or simple tokenization?
What form might property tokenization take? Are NFTs most promising? Simple tokenization? Or maybe some other forms, such as Soulbound tokens, or SBTs, as proposed recently by Vitalik Buterin?
Tamar says tokenization could assume a variety of forms. "Some will have a part of their cap table on Web3 (for decentralized finance); some will issue NFTs for timeshares and rentals; and others will use tokens for payments or asset management."
Like, sometimes people send me hitler coins to my public .eth address, and I don't have a way to stop this, but it doesn't really hurt me because etherscan doesn't show them unless you intentionally scroll all the way down the really long list of spam tokens. SBTs are similar.
— vitalik.eth (@VitalikButerin) May 12, 2022
Jarib Figueredo, a candidate for the State House of Representatives in the U.S., who joined Singh on the NFT.NYC 2022 panel on "Understanding the Value of Tokenized Real Estate," notes that "in Florida, there are issues transferring deeds, and these can be improved with NFTs — digital files that can prove ownership." Timesharing is another promising use case.
"In the future, real estate title could be issued as a nonfungible token," Black tells Magazine. "The owner of the property would maintain ownership of the token in a digital wallet. Changes in the value of the property would be reflected as changes in the value of the NFT." Further:
"When the owner wants to sell the property, they could list the property NFT on an NFT exchange. Buyers could bid on the NFT, and the NFT would be transferred to the digital wallet of the winning bidder. This would make the secondary market for real estate much more liquid and transparent."
Dilendorf, for his part, doesn't see NFTs or DAOs playing a dominant role in the real estate marketplace because they are essentially unregulated, unlike Reg CF-enabled digital securities, which are SEC-sanctioned.
Which kind of real estate projects are most ripe for tokenization? "Landmark assets will be the most successful," says Tamar, "and top-tier stadiums will fall under this category. Large institutional quality properties will be more likely to get tokenized, as they will experience more liquidity, and they will attract institutions during the primary sale unlike smaller properties."
Raising up the world's poor
Blockchain technology, too, can be of use in large swaths of the developing world where the existing infrastructure for ensuring property rights is weak or non-existent, Black tells Magazine, referencing Peruvian economist Hernando de Soto, who has argued that property rights are the key to economic growth. "If blockchain can improve property rights like real estate ownership in developing countries, this could be transformative for entire economies," says Black.
In an oft-cited Wall Street Journal opinion piece, de Soto emphasized blockchain's ease-of-access record-keeping that can be continuously updated as property ownership changes. Most of the world's population has no access at all "to a formal system of property rights," he wrote, adding:
"If Blockchain technology can empower public and private efforts to register property rights on a single computer platform, we can share the blessings of private-property registration with the whole world."
Global disrupter or niche player?
Can the crypto industry reach a point in the future where a significant proportion of real estate will be tokenized — e.g., more than 10% of the global real estate universe — or will this remain a niche area?
"I can imagine that day" — i.e., 10% tokenization — says Tawil, "but tokenization of real property ownership is likely going to take some time, especially in places like the United States where large infrastructural changes will be required."
Property deeds are recorded and stored in thousands of municipal offices, after all. These will have to be digitized. Then, too, there exists "a large lobby of professionals that profit from property transactions, which may be marginalized or eliminated in a tokenized property ownership world, such as lawyers, title insurance, brokers, etc.," Tawil tells Magazine.
According to Black, "Tokenization of 10% of the real estate market is not too difficult to imagine." Blockchain and NFTs provide a digital record of ownership that can be associated with physical assets, including real estate, autos "and any form of durable good."
"My prediction is that a large part of the property market will be tokenized to leverage the blockchain economy, but the top 20 percentile — by demand, size, performance, brand, location, etc. — will enjoy 80% of the financing opportunities," adds Tamar.
As with any disruptive technology, some business sectors could suffer if tokenized property catches on. Title insurance companies could be among the first to feel the pain. "Once there is a clear and transparent ledger that records who owns a property and whether there are any liens on the property, there will no longer be any justification for title company fees," Black tells Magazine. Still, success often begets success, or as Black puts it:
"As societies become more familiar with blockchain as an infrastructure for maintaining digital assets, this could open the door for applying some of these same principles to physical assets like real estate."
Source link


text:
Phosporus in the natural environment and the food chain has been a topic of several posts on my blog. So it didn't come as a surprise to find yet another diagram on phoshphorus flows over at Nels's MFA Diagram blog (one of the blogs I follow closely, see blogroll).
MFA diagrams have their focus on the nodes and the build-up of stocks. Sometimes they get a touch of Sankey diagram with the arrows having different magnitudes. The MFA diagram below is for phosphorous flows in China 2008 (original source: Min Qiao, Yuan-Ming Zheng, Yong-Guan Zhu, 2011. Material flow analysis of phosphorus through food consumption in two megacities in northern China). Values are in tonnes.
(click image to enlarge)
We can detect arrows with three different brush widths (my guess is 1px, 2px and 4 px), each standing for a value range into which the actual flow quantity falls. This may, however, bes somewhat misleading when having a quick glance at the diagram.
I quickly "translated" the above diagram to a Sankey diagram with flow values being actually to scale.
(click image to enlarge)
Here it is quite clear where the major phosphorus flows are located (from food production via urban consumption to sewage treatment plant and solid waste disposal: 2923 out of 5374 tons end up here). The other flows are comparatively small, with the phoshporous flow going directly to the aquatic system worth a mention. Two small flows in the center of the diagram are negligible, they are in fact so tiny in comparison to the major flows that they even don't show up (or just as a hairline) here.
I have therefore added a minimum width of 1 px for small flows so that the annual 17 tons from urban consumption and the 1.9 tons from rural consumption to the solid waste disposal are at least visible (albeit not to scale with the other flows any more).
(click image to enlarge)
Final phosphorous sinks are solid waste disposal (landfill?) and the aquatic system.


text:
Nature's Great Events
If you liked Planet Earth, you should probably check out Nature's Great Events. Narrated by David Attenborough and currently airing in the UK on BBC1 and BBC HD, the series consists of six 50-minute shows, each of which features a large-scale annual event, like the spring thaw in the Arctic Circle and the sardine run along the coast of South Africa. The series was shot in HD using many of the techniques seen in Planet Earth.
If you're in the UK, you can check out the first three episodes on the BBC site. In the US, Discovery will be airing the show sometime in the spring under the title Seasons of Survival (apparently Nature's Great Events isn't dramatic enough for the American audience). No word on whether Attenborough's expert narration will also be replaced as it was in Planet Earth.
In the meantime, some HD clips of the show are available on YouTube. This slo-mo video of a grizzly bear shaking the water off its fur is fun to watch but this too-short clip of an extraordinary coordinated attack of dolphins, seals, sharks, and birds on a massive school of sardines is the gem.
(via we made this, who call the series "mind-blowingly good")


text:
cutest DIY purses (with interchangable accessories)
As an Amazon Associate I earn from qualifying purchases.
(A no sew tutorial for the sweetest interchangeable girl's purses-and-accessories).
Pi is an accessory girl. She always has been, since she was teeny! And when we were packing for our trip to Europe earlier this year, she had some serious hopes for taking the cutest outfits with alltheaccessories. Since we were taking a total of two suitcases for a month long trip for five people (including a baby, whose travel crib and other baby stuff basically commandeered one whole suitcase), our available accessory space was limited at best.
These purses with interchangeable snap on accessories were the perfect compromise- they gave my accessory girl all sorts of options (without taking up too much space in our suitcase!).
In fact, she's continued using the purses even at home because of all of the coordinating goodness they offer!
To make these purses, I added snaps to the center of two neutral purses (the strong magnetic snaps). I made a couple of little slits with a razor blade for the prongs and then set the magnet in place.
Then I made a pile of accessories, also with magnetic snaps attached (be sure to use complimentary snap sides- all of the accessories with one snap side to fit into the snap you put on the purses).
-two bows (in fabric that matched two of Piper's dresses)
-one scarf (also in coordinating fabric- this one didn't have a snap, we just tied it onto the purse handle)
-a couple felt flowers
-a felt heart
-a giant pompom in Ravenclaw colors (I didn't take nearly enough pictures of all of our Harry Potter touring, but the Ravenclaw pompom was positively worn out at the end of the trip!)
After looking through these pictures I have a few final thoughts- first- I wish I'd taken more pictures in Europe- haha!
Also, I need to make a bunch more purse accessories for Piper for Christmas, and I definitely need to remake a Ravenclaw-ish accessory for my Harry Potter loving girl.
Related- this would make SUCH a great gift if you have an accessory girl!
ps- Even though most of my pics are of the straw purse, she totally used the brown bucket purse as well. 😉 Picture proof:
How to make a felt flower garland


text:
10" Inch White Round Cardboard Cake Board
Packs | Price per unit | Price inc GST |
Buy 1 - 25 QTY | $0.99 ea | $24.68 ea |
Buy 4 - 100 QTY | $0.78 ea | $78.00 ea |
The 10" Inch White Round Cardboard Cake Board is the perfect board to hold your handcrafted cake
These boards are very sturdy in design being 3mm, which is thicker than others. After spending hours baking a beautiful dessert cake, you can rest assured knowing this board will provide the stability you require. Presented in a stunning gloss white finish, the boards will remain clean and easily wipeable if you happen to drip icing on board. These cake boards are ideal for weddings, birthdays, corporate events, or for selling as a cake retailer. These cake boards are made from high-quality paperboard, 100% Food Grade Recycled Paper.
Diameter: 10" Inches (25.4cm)
Thickness: 3mm (0.12" Inches)
- 10" Inch White Round Cardboard Cake Board
- Stored and ready to send from our warehouse in Melbourne, Australia
- GST included
We use trusted transportation services like Direct Freight Express or StarTrack Express that deliver Australia wide to Melbourne, Sydney, Brisbane, Adelaide, Perth and even remote areas, we also deliver internationally! All of our prices include GST. These items are manufactured in China.


text:
The new technology, developed and extensively field-tested by ExxonMobil, more efficiently removes water vapor present during the production of natural gas. Removing water vapor through the use of dehydration technology, typically accomplished using large and expensive dehydration towers, reduces corrosion and equipment interference helping to ensure the safe and efficient transport of natural gas through the supply infrastructure and ultimately to consumers.
cMIST reduces the size, weight and cost of dehydration, resulting in reductions of surface footprint by 70 percent and the overall dehydration system's weight by half, which has significant added benefits on offshore applications.
"By leveraging our industry-leading experience with upstream applications, our researchers were able to create this advanced natural gas dehydration technology, which represents a step-change in operational efficiency and a significant reduction in footprint," said Tom Schuessler, president of ExxonMobil Upstream Research Company.
ExxonMobil's cMIST technology relies on a proprietary droplet generator to break up conventional solvent into tiny droplets that become well dispersed in the gas flow thereby increasing the surface area for the absorption of water from the gas. This is followed by an inline separator that coalesces the water-rich glycol droplets and moves them to the outside wall of the pipe for effective separation from the dehydrated natural gas. The water-rich glycol is regenerated using a conventional system and is sent back to the droplet generator to be used again. The droplet generator uses the energy from the flowing natural gas to create droplets of the right size.
ExxonMobil has licensed cMIST technology to the Chemtech division of Sulzer, a leading player in separation technologies, to facilitate deployment across the oil and gas industry.
"We are proud to have been selected as worldwide exclusive licensee of the cMIST technology, which includes our patented compact HiPer™ inline separator," said Torsten Wintergerste, president of the Chemtech division. "We look forward to servicing the oil and gas industry with this unique technology, allowing for much needed reductions in capital expenditures for both greenfield projects and existing facilities seeking brownfield debottlenecking opportunities. cMIST technology complements the Sulzer line of compact multi-phase separation technologies and will maximize benefits available to oil & gas operating companies around the world."
ExxonMobil's Houston-based Upstream Research Company and New Jersey-based ExxonMobil Research and Engineering division develop a range of innovative technologies aimed at producing energy more efficiently. The two organizations employ more than 1,000 PhD scientists and engineers.


text:
Rider Hotel Obereggen - for motorcyclists & active holidays in the Dolomites
Welcome to Rider Hotel & Steakhouse in Obereggen!
The Rider Hotel Obereggen right in the heart of the ski and hiking area Obereggen-Latemar is located in a natural playground for active holidays in the Dolomites - for young and old. Whether motorcyclists, mountain bikers, paragliders, skiers,cross-country skiers and especially snowboarders which will find their own Obereggen snow park - here you find the right home for your holidays in the Dolomites - UNESCO World Heritage! The Rider Hotel Rauth offers in an ideal location the right comfort and even more than you maybe expect expected at a fair price. Experience the Dolomites - UNESCO World Heritage - in a very special way!
Petra & Ferdi Weissensteiner
and the Rider Hotel Obereggen team


text:
The Bizarre Files
High school student arrested in robbery and shooting because he left his homework behind… Grandma threatened to use white witchery on a police officer… Man leads deputies on slow-speed chase in stolen forklift… & more
Kathy Romano presents "Her Story," a weekly tribute to women every Sunday morning at 7am on WMMR. She will introduce you to another woman making a major difference in the […]
Play Latest (July 12th)
118 episodes
Our Delaware Valley introduces you to the people and organizations making a difference in our community, from health and safety issues, to education, employment and even entertainment opportunities. Our Delaware […]
Play Latest (July 10th)
158 episodes
Weekday mornings on 93.3 WMMR Philadelphia with hilarious conversation, celebrity interviews & contests from people a little goofier than you.
Play Latest (July 24th)
2341 episodes
Conversations with WMMR's Pierre Robert, Jacky Bam Bam & Brent Porche.
Play Latest (May 21st)
1124 episodes
A favorite segment from our show…celebrity interviews and free-wheeling conversation appropriately sized for your busy day.
Play Latest (March 27th)
313 episodes


text:
HARTLEY v. COMMISSIONER OF INTERNAL REVENUE.
55 S.Ct. 756
79 L.Ed. 1399
No. 602.
Argued April 11, 12, 1935.
Decided April 29, 1935.
As Amended May 20, 1935.
Messrs. H. C. Fulton and H. B. Fryberger, both of Duluth, Minn., for petitioner.
The Attorney General and Mr. David E. Hudon, of Washington, D.C., for respondent.
Mr. Justice STONE delivered the opinion of the Court.
The Court of Appeals for the Eighth Circuit, 72 F.(2d) 352, affirmed a ruling of the Board of Tax Appeals, 27 B.T.A. 952, and held that under section 202 of the Revenue Act of 1921, c. 136, 42 Stat. 227, 229, and section 204 of the Revenue Act of 1924, c. 234, 43 Stat. 253, 258 (26 USCA § 935 and note), the basis for computing gain or loss on the sale of property, and its depletion or depreciation, for purposes of taxing income returned by the petitioner, an executor, is its value at the date of the decedent's death, rather than the cost to the decedent, or the value on March 1, 1913, if acquired before that date.
We granted certiorari, 294 U.S. 700, 55 S.Ct. 507, 79 L.Ed. -, to resolve a conflict of the decision below, and of the like decision, under section 202 of the 1918 Revenue Act, c. 18, 40 Stat. 1057, 1060 of the Circuit Court of Appeals for the Sixth Circuit in Eldredge v. United States, 31 F.(2d) 924, 930, with that of the Court of Claims in McKinney v. United States, 62 Ct.Cl. 180. See Elmhirst v. United States, 38 F.(2d) 915; Myers v. United States, 51 F.(2d) 145; compare McCann v. United States, 48 F.(2d) 446, each decided by the Court of Claims.
Petitioner's tax returns1 were for the calendar years 1924 and 1925. Sections 202(a, b) and 214(a)(8), (10) of the 1921 Act, 42 Stat. 229, 239—241, and section 204(a), (b), (c) of the 1924 and 1926 Acts (26 USCA § 935 and note) provide that the basis for computing gain or loss on the sale of property, and depreciation and depletion, shall be its cost, or its value on March 1, 1913, if acquired before that date. None of the acts specifically provide a basis for making the computations where return is made of income received by the estate of a decedent in the course of administration. But in the case of property acquired by 'bequest, devise, or inheritance' section 202(a)(3) of the 1921 Act, 42 Stat. 229, and section 204(a)(5) of the 1924 and 1926 Acts, 26 USCA § 935 and note, provide that the basis shall be the fair market value 'at the time of acquisition.'
The revenue acts consistently treat the estate of a decedent in the hands of an administrator or executor as a separate taxpayer. By section 2 of the 1921 Act, 42 Stat. 227, and 1924 and 1926 Acts, 26 USCA § 1262, the estate of a decedent is embraced within the term 'taxpayer.' Each act specifically provides for taxation of the income of an estate during administration. Section 219 of the 1924 Act, 42 Stat. 246, and 1924 and 1926 Acts, 26 USCA § 960 note. Each includes profits from the sale of property by the taxpayer in taxable income, section 213 of the 1921 Act, 42 Stat. 237, 238, and 1924 and 1926 Acts, 26 USCA § 954 and note, and provides for the deduction of losses from gross income in arriving at taxable income, section 214 of the 1921 Act, 42 Stat. 239, and 1924 and 1926 Acts, 26 USCA § 955 and note. See Merchants' Loan & Trust Co. v. Smietanka, 255 U.S. 509, 516, 517, 41 S.Ct. 386, 65 L.Ed. 751, 15 A.L.R. 1305. Each makes provision for the imposition of a tax upon the estates of deceased persons, and the 'gross estate' which is the basis for computing the tax is the value of the decedent's property at the time of his death. Section 402, 1921 Act, 42 Stat. 278; section 302, 1924 and 1926 Acts, 26 USCA § 1094 note.
The Court of Claims held that the time of acquisition, indicated by section 202(a) of the 1921 Act and section 204(a) of the 1924 and 1926 Acts as the controlling date for calculating gain or loss to the estate in the course of administration, must be taken to be the date of acquisition by the decedent rather than the time of acquisition by the executor or administrator on the decedent's death. This conclusion, it was thought, was compelled by the statutory command that the basis of computation shall be 'cost,' which could have no application to the acquisition by the executor or administrator, who is not a purchaser of the estate which be administers. McKinney v. United States, supra, 62 Ct.Cl. 180, 188. But this specification is not enough to restrict the effect of the general provisions of these acts which impose a tax on the income, including capital gains, of taxpayers. The use of the word cost does not preclude the computation and assessment of the taxable gains on the basis of the value of property, rather than its cost, where there is no purchase by the taxpayer, and thus no cost at the controlling date. See Heiner v. Tindle, 276 U.S. 582, 585, 586, 48 S.Ct. 326, 72 L.Ed. 714; Lucas v. Alexander, 279 U.S. 573, 578, 579, 49 S.Ct. 426, 73 L.Ed. 851, 61 A.L.R. 906.
No plausible reason has ben advanced for supposing that Congress intended the capital gains or losses of the estate of a decedent to be treated any differently from those resulting from the sale of property taken by 'bequest, devise, or inheritance,' as provided in section 202(a)(3) of the 1921 Act and section 204(a)(5) of the 1924 and 1926 Acts, or that it intended to bring gains or losses, accruing between the date of decedent's acquisition of the property and his death, into the computation of both the estate tax and the income tax assessed upon his administrator or executor. When it had a different purpose in the case of gifts inter vivos, not subject to a gift tax, it specifically directed that gains or losses to the donee should be computed on the basis of the cost of the property at the date of acquisition by the donor. Section 202(a)(2), 1921 Act; section 204(a)(2)(4), 1924 and 1926 Acts, 26 USCA § 935(a) (2, 4); see Taft v. Bowers, 278 U.S. 470, 49 S.Ct. 199, 73 L.Ed. 460, 64 A.L.R. 362; Helvering v. New York Trust Co., 292 U.S. 455, 462, 54 S.Ct. 806, 78 L.Ed. 1361. The conclusion seems inescapable that the intended date of acquisition by an executor or administrator, where the estate is the taxpayer, is the date of the decedent's death. Brewster v. Gage, 280 U.S. 327, 335, 50 S.Ct. 115, 74 L.Ed. 457.
Possibility of doubt was removed by Treasury Regulation. Article 343 of Regulation 45, under the 1918 Act, prescribed that gains or losses of an estate should be computed on the basis of the value of the property at the date of the decedent's death. This was carried forward by article 343 of Regulation 62 under the Act of 1921, of Regulation 62 under the Act of 1924, and of Regulation 69 under the Act of 1926. Following the decision of the Court of Claims in McKinney v. United States, supra, and with the purpose of conforming to it, the ruling was amended by T.D. 4011, VI—1 Cum. Bul. 77, on April 6, 1927, so as to make the cost to the decedent the basis of the computation. Doubts having been raised as to the ruling in McKinney v. United States, supra, by later decisions,2 the amendment was revoked, and article 343 restored to its original form, T.D. 4177, VII—2 Cum.Bul. 134, on July 7, 1928. The substance of the regulation in its original and final form was carried into § 113(a)(5) of the Revenue Act of 1928, c. 852, 45 Stat. 791, 818, 819, 26 USCA § 2113(a)(5), which directed that the basis for the computation of gains or losses upon property acquired by the decedent's estate from the decedent should be its value at the time of the decedent's death.
The re-enactment of the pertinent provisions of section 202 of the Revenue Act of 1921 (42 Stat. 229) in the Acts of 1924, 26 USCA § 935 and note, and 1926, 26 USCA § 935, without material change, was a congressional recognition and approval of the interpretation of the section by the Treasury Regulations, which gave them the force of law. Old Mission Portland Cement Co. v. Helvering, 293 U.S. 289, 293, 294, 55 S.Ct. 158, 79 L.Ed. 367; Brewster v. Gage, supra, 280 U.S. 327, 337, 50 S.Ct. 115, 74 L.Ed. 457. The incorporation of the regulation in section 113(a)(5) of the 1928 Act was intended to clarify, but not to change, the law. See Report of House Committee on Ways and Means, No. 2, 70th Cong., 1st Sess., p. 18; Report of Senate Committee on Finance, No. 960, 70th Cong., 1st Sess., p. 26; Report of Joint Committee on Internal Revenue Taxation, H.R. Doc. 139, 70th Cong., 1st Sess., pp. 17, 18.
Opinion modified in 295 U.S. 719, 55 S.Ct. 832, 79 L.Ed. -.
As permitted by section 702 of the Revenue Act of 1928, c. 852, 45 Stat. 791, 879, 26 USCA § 2702, petitioner elected to have his tax determined 'in accordance with the law properly applicable,' section 702(b), rather than 'in accordance with the regulations in force at the time such return was filed,' section 702(a).
Nichols v. United States, 64 Ct.Cl. 241; Bankers' Trust Co. v. Bowers, 23 F.(2d) 941 (D.C.S.D., N.Y.), In re Straight's Appeal, 7 B.T.A. 177.
The following state regulations pages link to this page.


text:
Buffy introduces songs from her new album "Medicine Songs".
This is a collection of front line songs about unity and resistance – some brand new and some classics – and I want to put them to work. These are songs I've been writing for over fifty years, and what troubles people today are still the same damn issues from 30-40-50 years ago: war, oppression, inequity, violence, rankism of all kinds, the pecking order, bullying, racketeering and systemic greed. Some of these songs come from the other side of that: positivity, common sense, romance, equity and enthusiasm for life.
I've found that a song can be more effective than a 400-page textbook. It's immediate and replicable, portable and efficient, easy to understand – and sometimes you can dance to it. Effective songs are shared, person-to-person, by artists and friends, as opposed to news stories that are marketed by the fellas who may own the town, the media, the company store and the mine. I hope you use these songs, share them, and that they inspire change and your own voice.
It might seem strange that along with the new ones, I re-recorded and updated some of these songs from the past using current technologies and new instrumentations – giving a new life to them from today's perspective. The thing is, some of these songs were too controversial for radio play when they first came out, so nobody ever heard them, and now is my chance to offer them to new generations of like-minded people dealing with these same concerns. It's like the play is the same but the actors are new.
I really want this collection of songs to be like medicine, to be of some help or encouragement, to maybe do some good. Songs can motivate you and advance your own ideas, encourage and support collaborations and be part of making change globally and at home. They do that for me and I hope this album can be positive and provide thoughts and remedies that rock your world and inspire new ideas of your own. ~ Buffy
Date and Venue TBA


text:
BERLIN – The Board of Zoning Appeals (BZA) granted the Berlin Fire Department (BFD) a variance this week, allowing the installation of a new LED sign on its property. However, the BZA did attach a few stipulations onto the approval.
Representatives of both the BFD and Selby Signs, the company that would be responsible for creating the new sign, addressed the BZA.
Berlin Superintendent of Planning and Zoning Chuck Ward informed the assembly that several criteria would have to be met for the fire company to qualify for a variance. Ward explained that the BFD would have to prove that there were extraordinary circumstances attached to the case, that property rights were not being fulfilled and that there would be no negative impact the property rights of neighbors.
Ward addressed the board on behalf of his department first, focusing on each of the criteria for a variance in order. According to Ward, there were no extraordinary circumstances in the case that would permit a variance. He also did not think that the original decision made by his department not to allow the installation of a new sign violated the BFD's property rights in any way.
"The applicant already enjoys full property rights," said Ward, pointing out that there was already a sign on the property that had served the BFD for years.
While the fire department does have the right to upgrade its sign, Ward explained that the proposed upgrade was in direct violation of town standards, as the proposed LED sign constituted "direct lighting," something not allowed for signs in Berlin.
Finally, Ward informed the BZA that he believed the sign might negatively impact surrounding properties. He mentioned that Main Street was considered a corridor into Berlin and held certain aesthetic standards because of the historic value of the street and the town.
"All development has to be subject to these standards," said Ward.
Additionally, Ward directed the council to a letter written by Eunice Sorin, a resident of the town who lives across the street from the fire department. Sorin expressed concern that the sign would diminish the historic quality of Berlin.
"This modern proposed flashing LED sign would be a detriment to the town and those that live and work here," Sorin wrote. "Once a special exemption is granted, it sets precedents for others to follow."
Selby argued that the town did allow direct lighting in some cases and wondered why the BFD was being singled out. Ward explained that the town did not allow direct lighting in signs or for advertising and those items like direct-lit holiday decorations didn't violate town standards.
After some debate over exactly how the property was zoned and a few hypothetical questions on how to define direct versus indirect lighting, BZA Chairman Woody Bunting informed Selby and Covington that they needed to stay on topic.
"There are three issues that you need to address with the board that you haven't touched on yet," he said.
At this point, BFD President David Fitzgerald approached the council to assist Selby and Covington. He made a counter presentation to each of Ward's concerns. Fitzgerald first asserted that there were special conditions involved with the case.
"We feel we have exceptional circumstances … there are no other fire companies that operate in that zone," said Fitzgerald.
He also believed that the current sign did not fulfill the BFD's property rights, at least compared to the new one, as the proposed sign would allow remote access, instant message change and automatic functions.
Finally, in response to the worries that the sign might negatively impact his neighbors, Fitzgerald stated that the sign could be kept to the same brightness level as the one currently in place. It could also be completely turned off late at night so as not to be a distraction.
After the case was heard, the BZA voted to grant the variation allowing the sign on a few conditions: it would be dimmed at sundown and switched off at midnight baring an emergency. Additionally, there would be a 15-second minimum delay between message changes to avoid "flashing." The last stipulation was that the new sign would not exceed the brightness of the old one.
Both the board and the BFD agreed on the conditions and the variance was passed.


text:
If you run a WordPress blog, drop everything and go check your blog security! Wait… I mean drop everything except reading this blog. Read, then drop. Right, hope that is sorted out now.
The point is that there is apparently a security hole in WordPress that is A) a serious pain in the ass, and B) nobody is quite sure how it is spreading. We were his with this on the Ignite Phoenix blog, and it took us the better part of a day to get back on our feet.
What It Is
We saw the problem when people visiting any of our pages were redirected to malware installation sites. People using Chrome were getting errors immediately, and it was triggering anti-virus software from AVG to Norton's. After some research and help from Chuck Reynolds, we determined he had a variant of this Cloaking Hack. Originally just adding keywords to the infected site, the new version we had put the redirect into every page on our site.
We tried to scrub it from our files and database (yes, it messes with your database) but we kept getting re-infected. We finally ended up having to move to a whole new ipage hosting and reinstall everything from the ground up. This was tricky since we could not reliably do an export from the previous site.
If you get it
Huge thanks to WP blogger's cloaking hack post for getting us started down the road to repair. If you're hit, I suggest starting here. This bugger also hit Chris Pearson, the developer of the Thesis Theme, who made a great post on diagnosing and repairing the pharma hack (our problem was a variation of this). The two of these are great resources and point to other places you can look to if you're hit.
Secure your blog!
Do not want! Trust me, take the time and do what you can to avoid this.
- Remove any old plugins and themes you're not using. Some hosting providers install a ton of themese by default. Get rid of them.
- Use strong passwords. Don't use something simple and easy to guess, or even common terms. This is good advice in general, but is is amazing how often it is ignored.
- Keep up to date on software versions. Don't let your plugins in or core installation lag behind.
- Run a database backup plug-in, like WP DB Backup. You can have it back up your database and email it to you every day/week. We had to use one of our backups to restore to when we rebuilt the site.
- Follow these WordPress security tips if possible. Some of these may be tricky for some people to do, but try all you can. #4 should be simple for everyone, and an absolute must.
- There are other good posts on WordPress security if you want to get deeper, including some recommended WordPress security plug-ins.
Take the time to review your site(s) and tighten them down. If one good things comes from our meltdown, let it be that we can help stop it from happening to someone else.
Related articles by Zemanta
- 9 WordPress Plugins to Assist With Supporting a Client's Website (kimwoodbridge.com)
- WordPress Security – A Comprehensive Guide (bloggingpro.com)
- JDI: Backup your WordPress (blogmistress.com)


text:
What to Give
All the ways to give what you can.
Gifts of Stocks and Bonds
A gift of your securities, including your stocks or bonds, is an easy way for you to make a gift. By making a gift of your appreciated securities, you can avoid paying capital gains tax that would otherwise be due if you sold these assets.
Gifts of Real Estate
A gift of your real property (such as your home, vacation property, vacant land, farmland or ranch, or commercial property) can make a great gift. If you own appreciated real property, you can avoid paying capital gains tax by making a gift to us.
Gifts of Retirement Assets
A gift of your retirement assets, such as a gift from your IRA, 401k, 403b, pension, or other tax deferred plan, is an excellent way to make a gift. By making a gift of your retirement assets, you will help further our work.
Gifts of Cash
A gift of cash is a simple and easy way for you to make a gift. You will receive a charitable tax deduction that will provide you with savings on this year's tax return.
Gift of Insurance
A gift of your life insurance policy is an excellent way to make a gift to us. If your life insurance policy is no longer needed or will no longer benefit your family, consider making a gift and help further our mission.
Non-cash Assets
The Greater New Orleans Foundation can accept an array of non-cash gifts, from closely held stock to personal property (such as art or jewelry).
To learn more about working with the Greater New Orleans Foundation, contact the Development team at email@example.com or call 504.598.4663.


text:
Chair of Theoretical Computer Science |
The main area of research in this unit is the design and the analysis of efficient algorithms and data structures, especially for geomtric problems. Typical examples are computational problems involving higher-dimensional polyhedra, localizing query points amidst a very large number of geometric objects, and approximative geometric query answering. As ancillary subjects we also study combinatorial geometry and randomization.
Currently the following more specific topics are being investigated: the efficient encoding of complicated geometric data structures such as triangulations, complexes, or arrangements; computations with complicated geometric objects such as ellipsoids; geometric algorithms in the ``transdichotomous model'', i.e. the exploitation of the limited parallelism provided by the usual computer word operations.
Raimund Seidel, Micha Sharir
Top-Down Analysis of Path Compression. pdf
SIAM Journal of Computing.
Raimund Seidel, Udo Adamy
On the Exact Query Complexity of Planar Point Location. pdf
Journal of Algorithms, pp. 189-217.
Markus Denny, Christian Sohler
Encoding a triangulation as a permutation of its point set. pdf
Proc. 9th Canad. Conf. Comput. Geom., pp. 39-43.
David Avis, David Bremner, Raimund Seidel
How good are convex hull algorithms? pdf
Comput. Geom. Theory and Appl., pp. 265-302.
Raimund Seidel, Cecilia Aragon
Randomized Search Trees. pdf
Algorithmica 16, pp. 464-497.
Raimund Seidel
A simple and fast incremental randomized algorithm for computing trapezoidal decompositions and for triangulating polygons. pdf
Comput. Geom. Theory and Appl., pp. 51-64.


text:
Sears Commits to Help New York Residents Move from Recovery to Rebuilding During the Aftermath of Hurricane Sandy
Continued Rebuilding Efforts Include Appliance Offers and Commitment to Rebuild Area Homes with Rebuilding Together
HOFFMAN ESTATES, Ill., Nov. 20, 2012 /PRNewswire/ -- Following a meeting with Governor Cuomo of New York, one of America's largest retailers, Sears, has committed its continued assistance with the East Coast rebuilding effort and the after-effects of Hurricane Sandy with programs that will bring much needed supplies and discounts on appliances to the New York area. Additionally, Sears along with its partner, non-profit Rebuilding Together, has made a commitment to rebuild 100 homes in the storm-affected areas.
(Photo: http://photos.prnewswire.com/prnh/20121120/CG17296)
"I personally reached out to Sears and asked them to help our community and they eagerly came up with programs that would be of immediate assistance," said Gov. Andrew Cuomo of New York. "I am challenging other retailers to please step-up and get involved."
Sears and Governor Cuomo discussed Sears' efforts to help residents of the region:
- Moving truck loads of essential goods including food, bottled water, batteries and blankets to the greater New York area to make sure necessities are on hand for those in need in the days right after the storm
- In the days right after the storm, providing SHOP YOUR WAY REWARDSM points to affected Members to help them in their rebuilding efforts
- Providing great appliance and home service values through coupons to Zone A residents
- Partnering with non-profit Rebuilding Together on an initial 10 home projects building up to 100 homes for those most impacted by the storm
"Immediately following the storm we sent truckloads of the most needed supplies and provided SHOP YOUR WAY REWARD points to Members impacted by the storm. It's clear people are hurting. Now Sears will help New York homeowners get back on their feet by helping them secure replacement appliances at great value," said Lou D'Ambrosio, Sears Holdings chief executive officer and president. "And our commitment with Rebuilding Together will help families regain a sense of normalcy in the upcoming months."
Sears, along with Rebuilding Together, will start to help rebuild immediately with a cluster of five to ten homes in one of the hardest hit communities in New York. The location will be announced in the upcoming days.
"An important step in overcoming a natural disaster's wrath is the rebuilding effort," said Gary Officer, CEO of Rebuilding Together. "We hope our commitment with Sears will inspire others to get involved with this very important process."
About Sears Holdings Corporation
Sears Holdings Corporation is a leading integrated retailer with over 2,600 full-line and specialty retail stores in the United States and Canada and the home of SHOP YOUR WAY, a social shopping experience where members have the ability to earn points and receive benefits across a wide variety of physical and digital formats through ShopYourWay.com. Sears Holdings is the leading home appliance retailer as well as a leader in tools, lawn and garden, fitness equipment and automotive repair and maintenance. Key proprietary brands include Kenmore, Craftsman and DieHard, with a broad apparel offering, including such well-known labels as Lands' End, the Kardashian Kollection, Jaclyn Smith and Joe Boxer, as well as Sofia by Sofia Vergara and The Country Living Home Collection. We are the nation's largest provider of home services, with more than 15 million service and installation calls made annually and have a long-established commitment to those who serve in the military through initiatives like the Heroes at Home program. We have been named the 2011 Mobile Retailer of the Year, Recipient of the 2012 ENERGY STAR® "Corporate Commitment Award" for Product Retailing and Energy Management and one of the Top 20 Best Places to Work for Recent Grads. Sears Holdings Corporation operates through its subsidiaries, including Sears, Roebuck and Co. and Kmart Corporation. For more information, visit Sears Holdings' website at www.searsholdings.com. Twitter: @searsholdings | Facebook: http://www.facebook.com/SHCCareers
About Rebuilding Together
Rebuilding Together believes that everyone deserves to live in a Safe and Healthy Home and that Community starts at Home. Our focus provides critical repairs, accessibility modifications and energy efficient upgrades to low-income homes and community centers at no cost to service recipients. Our impact extends beyond the individuals served to revitalize and stabilize vulnerable neighborhoods and communities across the country. Our 200 local affiliates complete 10,000 rebuild projects a year thanks to the efforts of nearly 200,000 volunteers from corporate partners, skilled trades professionals and everyday good citizens. Join us — visit www.RebuildingTogether.org
Tom Aiello | Danny Cohn |
Sears Holdings | Zeno Group |
email@example.com | email@example.com |
SOURCE Sears Holdings Corporation