GMS VECTORS REVISION Multiple Choice 91 If = 9i – 4j and = 5i – 4j, the magnitude of is equal to A 5 B 4 C 3 D 4 E 5 92 If a = 8i – 6j, then the unit vector in the direction of a is A 10 B 2 C (8i + 6j) D (8i 6j) E (6i 8j) 93 The vector has a magnitude of 12 and makes an angle of 60 measured anticlockwise from the positive x axis. When expressed in terms of i and j, would be written as A 6i + 6j B 6i – 6j C 6i + 6 j D 6i 6 j E 12i + 6 j 94 If a = i 2j + 3k and b = 2j 5k, then 2a + b is equal to A 6j + 11k B 2i 6j + 11k C 2i 4j 2k D 2i 2j + k E i + 8k 95 = i + 3j and = 4i 2j. If = , is the vector A 2i 4j B 3i + j C 2i + 8j D 4i 4j E 7i j 96 If a = i 2j + 3k and b = 2j 5k, then 2a b is equal to A 6j + 11k B 2i 6j + 11k C 2i 4j 2k D 2i 6j + k E i + 8k 97 = i 3j and = 4i 2j. If = , is the vector A 2i 4j B 3i + j C 5i 5j
b i ii c a + b + c d a b + 2c e 2(a b) + c f a + b 94 Find the vector b in the direction of vector a = i + 2j 2k such that b= 5. 95 In triangle OAB, = 6i + j and = 3i 2j. If M is the midpoint of AB, find . 96 OABCDEFG is a cuboid. = 4j, = 2i and = 2k. Express the following vectors in terms of i, j and k. a b c d e f g h Analysis 18 ABCDEF is a regular hexagon. a Find the diagonal BF in terms of a and b. b i Find AC in terms of a and b. ii find BC in terms of a and b. c Draw a diagonal vector v equal to a + 2b. d Draw a diagonal vector w to 2a + 2b. 19 W, T, R and V are the midpoints of SP, PQ, QX and SX respectively. Let = p, and = x. a Find in terms of p and r. b Find in terms of p, r and x. c Find in terms of p, r and x. d Find in terms of p and r. e What conclusion can you reach about the quadrilateral WTRV? Give reason(s) for your answer. 20 a The position vectors of A, B and C relative to an origin O are –i + pj, 5i + 9j and 6i + 8j respectively. i Determine the of p for which A, B and C are collinear. ii Given that D is a point on OC such that is a unit vector, find the position vector of D relative to O. b
Points P and Q have position vectors p and q respectively, relative to the O. The point R on OP is such that OR = OP. The point S on RQ is such that RS = RQ. i Express and in terms of p and q. PS produced meets OQ at T, where OT =  OQ. ii Express in of , p and q, and evaluate  21 The position of a boat at point B is given by the vector = 2i – j. The position of a light house at A is given by = 3i + j. (Distances are measured in kilometres.) a i Find . ii Find ||. iii If i is a unit vector in the east direction and j a unit vector in the north direction, find the bearing of B from A. b The boat now moves at a speed of 4 km/h in an easterly direction. Find its position relative to O after i 3 hours ii t hours c Find the distance of the boat from the lighthouse after t hours. ANSWERS: MC: 91 D 92 D 93 C 94 D 95 C 96 B 97 A 98 D 99 E 100 C SA: 90 a b (–3i j – 5k) 91 a i a + b = 3i + j ii a b = i 7j iii = (2i 3j) 92 a = (i + k) b = a + b c = = 93 a i ii 3 b i (i + j) ii (i + 2j 2k) c 4i + 2j 2k d i + 6j 4k e i + 6j 2k
f 3 94 b = (i + 2j 2k) 95 = i j 96 a 2i + 4j b 2i + 4j + 2k c 2i + 2k d 4j + 2k e 4j f 2i 4j g 4j + 2k h 4j + 2k Analysis: 18 a b a b i = b + 2a ii = a + b c, d 19 a p + r b 2p + 2r + 2x c p + r + x d p + r e Parallelogram; = 20 a i p = 15 ii = i + j b i = p + q = p + q ii = p + q = 21 a i = –i – 2j ii = iii 207 b i 14i j ii (2 + 4t)i j c km