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 ? q p – q p – q

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. An Important technique To establish or express the co-linearity of three points [Lie in a straight line] Choose any two line segments, AB, AC or BC. For the points to be co-linear AB, AC or BC must lie in the same direction Example Given OA = p, OC = q and OB = 2p – q , show that A, B and C are co-linear. AB = AO + OB = – p + (2p – q) = p – q B A C BC = BO + OC = – (2p – q) + q = 2q – 2p = –2(p – q ) = –2AB Hence A , B and C are co-linear. O AB &BC are parallel (even though in opposite directions) and have a common point B

12. 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.

13. The diagram shows quadrilateral OABC.OA = a, OC= c and OB = 2a + c (a) Find expressions, in terms of a and c, for (i) AB (ii) CB (iii) What kind of quadrilateral is OABC? Give a reason for your answer. (b) Point P lies on AC and AP = AC. (i) Find an expression for OP in terms of a and c. Write your answer in its simplest form. (ii) Describe, as fully as possible, the position of P. a) AB = AO + OB = a + c (ii) CB = CO + OB P = 2a Trapezium : CB is parallel to OA. b) OP = OA + AP =  a + c (ii) OB = 3 x (OP) They are parallel and have a common point, hence O, P & B are co-linear.

14. Example OACB is a parallelogram with OA = a and OB = b M is the midpoint of AC P is the intersection of OM with AB Obtain the position vector of M Given that AP = kAB use the triangle OAP to obtain an expression for OP in terms of a, b and k. Deduce the position vector P.

15. Example OACB is a parallelogram with OA = a and OB = b M is the midpoint of AC P is the intersection of OM with AB Obtain the position vector of M Given that AP = kAB use the triangle OAP to obtain an expression for OP in terms of a, b and k. Deduce the position vector P. A C OB + BM OM = = b +  a a AP = kAB OP = OA + AP O B = a + kAB b = a + k(b – a) = (1 – k)a +kb (iii) OP = hOM OP = h(b +  a) M P a 1 – k =  h bk = h Hence 1 =  hh =  OP =  a +  b