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

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

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

4. 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.

5. A few examples
b – a  a a b b

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

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

8. 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

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

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

11. 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)

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

13. 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

14. 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.

15. 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

16. 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