Vector equations of a line
Lines can be written in several different ways:
Vector equations of a plane
- 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:
- The vector product of any 2 parallel vectors is zero
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.
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.
- 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
- 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
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.
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:
- Proceed as for and intersection of a line and a plane
Distance between a point A, from a line L
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:
Distance between 2 skew lines
- 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