# finding parametric & symmetric equations of line

• Jun 8th 2008, 01:26 AM
Craka
finding parametric & symmetric equations of line
Have a question which is as above for line through points (0, 1/2, 1) and (2,1, -3)

They way I understand it this is what you do. Get a directional vector between the two points ie let (0, 1/2, 1) be point A and (2, 1, -3) be point B. Thus vector AtoB is < (2-0), (1-1/2), (-3-1)> giving < 2, 1/2, -4>

Now parametrics would be x= (0+2t) , y=( 1/2+1/2t) and z = (1+ (-4t))

giving symetric as being x/2 = (2y-1) = (z-1)/-4

• Jun 8th 2008, 07:08 AM
Soroban
Hello, Craka!

Quote:

Find the equations for line through points $(0,\frac{1}{2}, 1)\text{ and }(2,1, -3)$

The way I understand it this is what you do.
Get a directional vector between the two points $A(0,\frac{1}{2}, 1)\text{ and }B(2, 1, -3)$
Thus: $\vec{v} \:=\:\overrightarrow{AB} \:=\:\langle 2-0,\:1-\frac{1}{2},\:-3-1\rangle \:=\:\langle 2,\:\frac{1}{2},\:-4\rangle$

The parametrics would be: . $\begin{Bmatrix} x&=& 0+2t \\ y& =& \frac{1}{2}+\frac{1}{2}t \\ z &=& 1-4t \end{Bmatrix}$

giving symmetric as: . $\frac{x}{2} \;=\; \frac{2y-1}{1} \;= \;\frac{z-1}{-4}$

But it looks like I'm wrong . . . . WHO said you're wrong?

There is an unavoidable problem with the equations of a line.
. . I wish textbook authors and math teachers would be own up to it.

There are a zillion possible forms of the equations of a line.
. . There is no "one right answer."

You used point $A(0,\frac{1}{2},1)$ and $\vec{v} = \langle 2,\frac{1}{2},1\rangle$ . . . and your answers are correct.

Suppose you used $B(2,1,-3)\text{ and }\vec{v} = \langle 2,\frac{1}{2},1\rangle$
You would have: . $\begin{Bmatrix} x &=& 2 + 2t \\ y &=& 1 + \frac{1}{2}t \\ z &=& \text{-}3 -4t \end{Bmatrix}\;\;\text{ and }\;\;\frac{x-2}{2} \:=\:\frac{2y-2}{1} \:=\:\frac{z+3}{-4}$ . . also correct.

$\text{Instead of }\,\vec{v} = \langle 2,\frac{1}{2},-4\rangle$, we can use a parallel vector: . $2\vec{v} = \langle 4,1,-8\rangle$

With point $B(2,1,-3)$, we would have: . $\begin{Bmatrix}x &=& 2 +4t \\ y &=& 1 + t \\ z&=& \text{-}3 -8t \end{Bmatrix}\;\;\text{ and }\;\;\frac{x-2}{4} \:=\:\frac{y-1}{1} \:=\:\frac{z+3}{-8}$

You see, the variations are endless . . .

• Jun 8th 2008, 07:22 AM
Craka
Thankyou. That was causing me alot of anguish, as all the different sources I looked at seemed to have the method I was using, however the answer in the rear of my book was different to what I was getting.
The text passage contains a mathematical question and a detailed solution provided by the user, followed by a response from another user validating the solution and explaining the variability in the answers.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
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- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# finding parametric & symmetric equations of line

• Jun 8th 2008, 01:26 AM
Craka
finding parametric & symmetric equations of line
Have a question which is as above for line through points (0, 1/2, 1) and (2,1, -3)

They way I understand it this is what you do. Get a directional vector between the two points ie let (0, 1/2, 1) be point A and (2, 1, -3) be point B. Thus vector AtoB is < (2-0), (1-1/2), (-3-1)> giving < 2, 1/2, -4>

Now parametrics would be x= (0+2t) , y=( 1/2+1/2t) and z = (1+ (-4t))

giving symetric as being x/2 = (2y-1) = (z-1)/-4

• Jun 8th 2008, 07:08 AM
Soroban
Hello, Craka!

Quote:

Find the equations for line through points $(0,\frac{1}{2}, 1)\text{ and }(2,1, -3)$

The way I understand it this is what you do.
Get a directional vector between the two points $A(0,\frac{1}{2}, 1)\text{ and }B(2, 1, -3)$
Thus: $\vec{v} \:=\:\overrightarrow{AB} \:=\:\langle 2-0,\:1-\frac{1}{2},\:-3-1\rangle \:=\:\langle 2,\:\frac{1}{2},\:-4\rangle$

The parametrics would be: . $\begin{Bmatrix} x&=& 0+2t \\ y& =& \frac{1}{2}+\frac{1}{2}t \\ z &=& 1-4t \end{Bmatrix}$

giving symmetric as: . $\frac{x}{2} \;=\; \frac{2y-1}{1} \;= \;\frac{z-1}{-4}$

But it looks like I'm wrong . . . . WHO said you're wrong?

There is an unavoidable problem with the equations of a line.
. . I wish textbook authors and math teachers would be own up to it.

There are a zillion possible forms of the equations of a line.
. . There is no "one right answer."

You used point $A(0,\frac{1}{2},1)$ and $\vec{v} = \langle 2,\frac{1}{2},1\rangle$ . . . and your answers are correct.

Suppose you used $B(2,1,-3)\text{ and }\vec{v} = \langle 2,\frac{1}{2},1\rangle$
You would have: . $\begin{Bmatrix} x &=& 2 + 2t \\ y &=& 1 + \frac{1}{2}t \\ z &=& \text{-}3 -4t \end{Bmatrix}\;\;\text{ and }\;\;\frac{x-2}{2} \:=\:\frac{2y-2}{1} \:=\:\frac{z+3}{-4}$ . . also correct.

$\text{Instead of }\,\vec{v} = \langle 2,\frac{1}{2},-4\rangle$, we can use a parallel vector: . $2\vec{v} = \langle 4,1,-8\rangle$

With point $B(2,1,-3)$, we would have: . $\begin{Bmatrix}x &=& 2 +4t \\ y &=& 1 + t \\ z&=& \text{-}3 -8t \end{Bmatrix}\;\;\text{ and }\;\;\frac{x-2}{4} \:=\:\frac{y-1}{1} \:=\:\frac{z+3}{-8}$

You see, the variations are endless . . .

• Jun 8th 2008, 07:22 AM
Craka
Thankyou. That was causing me alot of anguish, as all the different sources I looked at seemed to have the method I was using, however the answer in the rear of my book was different to what I was getting.
user
The text passage contains a mathematical question and a detailed solution provided by the user, followed by a response from another user validating the solution and explaining the variability in the answers.

TYPE: SINGLE_QUESTION_WITH_ANSWER