Addition of vectors

1. (i) Triangle Rule [For vectors with a common point] C Addition of vectors 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 r p+q+r q p

4. 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. Which vector is represented by p + q ? • Which vector is represented by p – q ? q p p p + q q – q p – q p

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

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

12. P A a N O b B Q OA = AP and BQ = 3OB a) 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 B A n m+n m F C D E

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? BC = BO + OC a) = 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. A C D a E O b B Example In the diagram, and , OC = CA, OB = BE and BD : DAhas 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.

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