• # Revision:Further vectors

TSR Wiki > Study Help > Subjects and Revision > Revision Notes > Mathematics > Further Vectors

## Vector equations of a line

Lines can be written in several different ways:

There’s the c4 way:

The other c4 way:

Also the Cartesian form:

## Vector equations of a plane

Similar to the c4 line equation:

Similar again: I used numbers for the sake of notation. 2 i’s won’t look pretty!

Here’s a new one:

Or:

Or:

## Vector product

• The definition of the vector product is where is the unit vector perpendicular to both a and b in the direction given by the right hand screw rule.
• For vectors in component form, the vector product may be calculated using either of the following determinants, the second of which is given in the formula booklet:

or

• The vector product of any 2 parallel vectors is zero
• The vector product is anti-commutative as
• Area of a triangle is , of a parallelogram is , and representing adjacent sides.

## Intersections

#### Point of intersection of a line and a plane

• Express the position vector of a general point on the line as a single vector, e.g.

• substitute this vector into the normal form of the equation of a plane and find , e.g.

etc.

• Substitute your value of back into the equation of the line.

#### Line of intersection of 2 intersecting planes

• Find the direction vector of the line by using the fact that the line is perpendicular to both normal vectors, i.e.
• Find any point , common to both planes by using any value (usually 0) for x, y or z
• Equation of line is

## Angles

• The direction vector of a line is given by d in the equation r= d
• The direction of the normal to a plane is given by n in the equation r.n = a.n
• The angle between two planes is the angle between their 2 normals, n_1 and n_2. This is usually found by using the scalar product, a.b =
• The angle between two lines is the angle between their 2 direction vectors, d_1 and d_2.
• The angle between a line and a plane is found by finding the angle between d and n then subtracting from 90 degrees. Alternatively, if is the angle between d and n then as you may use

## Distances

• The distance of a plane r.n = p from the origin is or where the sign is unimportant.
• To find the distance between two planes, first find their distances from the origin. If the two p-values are the same sign, the planes are on the same side of the origin so subtract the 2 distances. If the two p-values are different signs, the origin lies between them so add the positive distances.

#### Distance between a plane and a point, A

1)*Express the equation in the form

• If the point is , quote the result from the formula booklet giving:

Distance =

2)*Find the equation of the plane through the given point A, parallel to the original plane, using r.n = a.n

• Find the distance between these 2 planes.

3)*Locate any point, P on the plane and find the vector

• If is the angle between and n, then the required distance h is so h =

4)*Find the co-ordinates of F, the foot of the perpendicular from A to the plane and find AF. This method should only really be used if F is specifically asked for because it’s a bit long winded. Anyway, F can be found as follows:

• As AF is perpendicular to the plane, then n, so = a + n
• Proceed as for and intersection of a line and a plane

#### Distance between a point A, from a line L

1)*Locate any point, P on L and find the vector

• If is the angle between and d, then the required distance h is so h =

2)* Find the co-ordinates of F, the foot of the perpendicular from A to L and find AF. F can be found as follows:

• As F lies on L, express the position vector of F as a single vector involving
• Use the fact that to find
• As is perpendicular to L, then so use this to find and hence F

#### Distance between 2 skew lines

• Locate any point, A on and any point B on and find the vector = ba
• Find the common normal, i.e. the vector which is perpendicular to both and . This is given by n = and may be simplified if appropriate
• If is the angle between and n, then the required distance h is so h =
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The text passage is a set of revision notes on further vectors in mathematics, covering topics such as vector equations of lines and planes, vector products, intersections, angles, and distances. It does not contain any specific mathematical questions or answers to questions.
no_question
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First, describe the content structure you found in 1-2 sentences.

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• # Revision:Further vectors

TSR Wiki > Study Help > Subjects and Revision > Revision Notes > Mathematics > Further Vectors

## Vector equations of a line

Lines can be written in several different ways:

There’s the c4 way:

The other c4 way:

Also the Cartesian form:

## Vector equations of a plane

Similar to the c4 line equation:

Similar again: I used numbers for the sake of notation. 2 i’s won’t look pretty!

Here’s a new one:

Or:

Or:

## Vector product

• The definition of the vector product is where is the unit vector perpendicular to both a and b in the direction given by the right hand screw rule.
• For vectors in component form, the vector product may be calculated using either of the following determinants, the second of which is given in the formula booklet:

or

• The vector product of any 2 parallel vectors is zero
• The vector product is anti-commutative as
• Area of a triangle is , of a parallelogram is , and representing adjacent sides.

## Intersections

#### Point of intersection of a line and a plane

• Express the position vector of a general point on the line as a single vector, e.g.

• substitute this vector into the normal form of the equation of a plane and find , e.g.

etc.

• Substitute your value of back into the equation of the line.

#### Line of intersection of 2 intersecting planes

• Find the direction vector of the line by using the fact that the line is perpendicular to both normal vectors, i.e.
• Find any point , common to both planes by using any value (usually 0) for x, y or z
• Equation of line is

## Angles

• The direction vector of a line is given by d in the equation r= d
• The direction of the normal to a plane is given by n in the equation r.n = a.n
• The angle between two planes is the angle between their 2 normals, n_1 and n_2. This is usually found by using the scalar product, a.b =
• The angle between two lines is the angle between their 2 direction vectors, d_1 and d_2.
• The angle between a line and a plane is found by finding the angle between d and n then subtracting from 90 degrees. Alternatively, if is the angle between d and n then as you may use

## Distances

• The distance of a plane r.n = p from the origin is or where the sign is unimportant.
• To find the distance between two planes, first find their distances from the origin. If the two p-values are the same sign, the planes are on the same side of the origin so subtract the 2 distances. If the two p-values are different signs, the origin lies between them so add the positive distances.

#### Distance between a plane and a point, A

1)*Express the equation in the form

• If the point is , quote the result from the formula booklet giving:

Distance =

2)*Find the equation of the plane through the given point A, parallel to the original plane, using r.n = a.n

• Find the distance between these 2 planes.

3)*Locate any point, P on the plane and find the vector

• If is the angle between and n, then the required distance h is so h =

4)*Find the co-ordinates of F, the foot of the perpendicular from A to the plane and find AF. This method should only really be used if F is specifically asked for because it’s a bit long winded. Anyway, F can be found as follows:

• As AF is perpendicular to the plane, then n, so = a + n
• Proceed as for and intersection of a line and a plane

#### Distance between a point A, from a line L

1)*Locate any point, P on L and find the vector

• If is the angle between and d, then the required distance h is so h =

2)* Find the co-ordinates of F, the foot of the perpendicular from A to L and find AF. F can be found as follows:

• As F lies on L, express the position vector of F as a single vector involving
• Use the fact that to find
• As is perpendicular to L, then so use this to find and hence F

#### Distance between 2 skew lines

• Locate any point, A on and any point B on and find the vector = ba
• Find the common normal, i.e. the vector which is perpendicular to both and . This is given by n = and may be simplified if appropriate
• If is the angle between and n, then the required distance h is so h =
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The text passage is a set of revision notes on further vectors in mathematics, covering topics such as vector equations of lines and planes, vector products, intersections, angles, and distances. It does not contain any specific mathematical questions or answers to questions.

TYPE: NO_QUESTION