text: • # 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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