C4 – Vectors Summary
 A vector has magnitude and direction, a scalar has magnitude only.
 In typed text vectors are printed in bold p or written as
OP , in
OP .
 Vectors may be written in component form as a  a1 i  a 2 j  a3 k or
handwriting vectors are underlined p or written as
 a1 
 
as a column vector a   a 2  .
a 
 3
PF is perpendicular to the line, i.e.
PF  b  0 . Also PF  OF  OP  (a   b)  OP . Use these to find

a  a1  a2  a3 .
2
2
2
 A position vector starts at the origin.
 A unit vector has magnitude 1, e.g. i and j.
 Vector  scalar: same direction, magnitude changes (multiply each
component by the scalar)
 Negative vector: opposite direction, same magnitude (change signs)
 Adding vectors: draw “nose to tail” to find the resultant (add each
component)
AB  OB  OA .
 The midpoint M of a line AB is
OM  12 (OA  OB ) .
 Points are collinear if they lie on a straight line.
 If points A, B and C are collinear then AB and BC are parallel and
share a common point B.
 The scalar/dot product of a and b is a  b  a1b1  a2 b2  a3b3 .
 The angle between 2 vectors is given by
cos  
ab
(Vectors
ab
should start from the same point.)
 2 vectors are perpendicular if a b  0 .
 Vector equation of a line:
a point on the line and
equations for each component and solve simultaneously).
iii) skew if not i) or ii) (in 3D only).
 The angle between lines is given by the angle between their directions.
 To find the distance between a point P and a line r  a   b , let F be
a point on the line such that
 The magnitude (size/length) of a vector is
 Vector joining 2 points:
r  a  b and r  c   d are
i) parallel if b  md , where m is a scalar (i.e. same direction).
ii) intersect if  and  exist for which a  b  c   d (write out
 2 lines given by
r  a   b where a is the position vector of
b is the direction of the line
 Equation of a line through A and B:
r  OA   AB .
then
PF . (F is sometimes called the foot of the perpendicular from
the point to the line.)