Revision:Further Vectors

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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 not 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 = b – a

• 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 =