all questions with full solutions. Make sure your work is legible, even after you have scanned it, and submit it as a single file. 1. Given a = [4, -2, 3] and b = [5, 6, -1], find a. b. ab c. d. The angle between a and b A vector perpendicular to a A unit vector in the direction of b 2. A force F = [3, -5, 2] in Newtons, pulls a sled through a displacement s = [5, -1, 3] in metres. a. How much work is done on the sled by the force? b. What is the minimum magnitude of force that could have been applied to the sled to obtain the same displacement? Explain your answer. 3. Given a = [-5, 8, 1] and b = [2, -7, -3], find a. a b and verify that it is perpendicular to both a and b b. A vector c such that a (b c ) = 0 What is the relationship between the vectors a , b and c in this case, and why? Verify this. 4. Given v = [3, -6, 2] and w = [-1, 6, 5], find; a. v w b. c. wv d. What does the direction of w v depend on What does the magnitude of w v depend on? 5. Draw diagrams to explain the answers to the following questions. a. Is it possible to have a b = 0? b. Is it possible to have a b undefined? c. Is it possible to have a b = b a ? d. Explain why a c = a (b c ) 6. Answer the following with either an explanation, a diagram or a proof: a. If a b = a c , what is the relationship between b and c ? b. If a b = a c , what is the relationship between b and c ? 7. Prove that a b = -(b a ) for all a , bER3 8. Given vectors a , b andc , state whether each of the following results in a scalar, a vector, or is not possible. Justify each response. a. a (b c ) b. c. ( a b ) c ( a b ) + (c d ) d. e. ( a b ) + (c d ) ( a b ) (c d ) f. ( a b ) (c d ) A blimp is held in place with two ropes. The wind is creating a force of 300 N on a bearing of 280 o and one of the ropes is exerting a force of 250 N on a bearing of 170 o. What force is being exerted by the other rope? 9. A pilot wishes to fly from Bayfield to Toronto, a distance of 195 km on a bearing of 085. The speed of the plane in still air is 260 km/h. A 20 km/h wind is blowing on a bearing of 215. Remembering that she must fly on a bearing of 085 relative to the ground (i.e. the resultant must be on that bearing), find a. The heading she should take to reach her destination. b. How long the trip will take