Question

You and a friend both purchase dune buggies. These off-road vehicles are excellent for driving on sand. To test out your new rides, you decide to play a game of cat and mouse in a desert with sand dunes. Both of you start in the same location. You give your friend a head start, then attempt to catch up teo them. To give you a chance, they radio you their coordinates several times You may assume that both dune buggies start at the origin (0,0,0), and that each coordinate contains an East-West component, a North-South component and a height component, relative to the origin. Let positive movement in the i direction be East, positive movement in the j direction be North, and positive movement in the k direction correspond to increasing height. Your friend radios you the following four coordinates in order O (0,0,0) P (250, 400, 30) P2 (800, 500, 20) Ps (1000, 800, 40) Note that the first transmission is the origin, just to test that your radios are working! (a) Determine the displacement vectors describing your friends trajectory between each of their trans- missions [1 mark (b) Determine the magnitude of the displacement of Ps from the origin [1 mark] (c) Suppose that when you receive the transmission corresponding to P2, you are located at a point Y with coordinates Y (500, 600). For this part, you may neglect the height component of all points, and consider the problem in terms of 2-dimensional vectors (East-West and North-South) (i) Determine the angle between your effective straight-line trajectory to the point Y and your friends effective straight-line trajectory to the point P2 1 mark] (ii) Suppose that you now travel from Y, parallel to P2Ps, such that if you were to continue on this tra- jectory without changing direction, you would eventually reach the point (1100,1500. Draw a schematic (without specific values) showing at least the points Y, Ps, your trajectory, and the point P on your trajectory that is closest to Ps [1 mark Hint: The shortest distance between a point and a line is on a path perpendicular to the line. ii Show that the closest that you get to the point Ps along this trajectory is (approximately) the point (746,969). [1 mark]

Answer 1

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