Addition of vectors (i) Triangle Rule [For vectors with a common point] C B A

(ii) Parallelogram Rule [for vectors with same initial point] D C B A

(iii) Extensions follow to three or more vectors p+q+r q p

First we need to understand what is meant by the vector – a Subtraction First we need to understand what is meant by the vector – a a – a a and – a are vectors of the same magnitude, are parallel, but act in opposite senses.

A few examples b – a  a a b b

q p p q – q p Which vector is represented by p + q ?

B CB = CA + AB = - AC + AB = AB – AC C A

OP is a position vector of a point P. We usually associate p with OP Position Vectors Relative to a fixed point O [origin] the position of a Point P in space is uniquely determined by OP P p OP is a position vector of a point P. We usually associate p with OP O

A very Important result! B AB = b - a b A a O

The Midpoint of AB A M OM = ½(b + a) a B b O

Vectors Questions P A a N O b B Q Example In the diagram, OA = AP and BQ = 3OB. N is the midpoint of PQ. and A P N Q O B a b Express each of the following vectors in terms of a, b or a and b. (a) (b) (c) (d) (e) (f) (g) (h)

OA = AP and BQ = 3OB a) A P N Q O B a b b) c) d) e) f) g) h)

Example ABCDEF is a regular hexagon with , representing the vector m and , representing the vector n. Find the vector representing m A B C D F E n m+n m

Example M, N, P and Q are the mid-points of OA, OB, AC and BC. OA = a, OB = b, OC = c (a) Find, in terms of a, b and c expressions for (i) BC (ii) NQ (iii) MP (b) What can you deduce about the quadrilateral MNQP? a) BC = BO + OC = c – b (ii) NQ = NB + BQ b =  c a (ii) MP = MA + AP c =  c MNPQ is a parallelogram as NQ and MP are equal and parallel.

Example In the diagram, and , OC = CA, OB = BE and BD : DA has ratio 1 : 2. a) Express in terms of a and b (i) b) Explain why points C, D and E lie on a straight line. O D B C E A b a

Example ABC is a triangle with D the midpoint of BC and E a point on AC such that AE : EC = 2 : 1. Prove that the sum of the vectors , , is parallel to