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Vikram starts from Point X and drives 8 km towards East. He then takes a left turn, drives 5 km, turns left and drives 8 km. He then takes a right turn and drives 3 km to stop at Point Y. How far (shortest distance) and towards which direction should he drive to reach Point X again?
8 km, West
8 km, South
8 km, North
5 km, West
8 km, South
Track the net displacement using a coordinate system (X, Y) where East is +x and North is +y; then find the vector from the final point back to the origin.
Track the net displacement using a coordinate system (X, Y) where East is +x and North is +y; then find the vector from the final point back to the origin.
Mapping the path with Cartesian Coordinates
Start at Origin (0,0). Move 8 km East: (8, 0). Left turn (North) for 5 km: (8, 5). Left turn (West) for 8 km: (8-8, 5) = (0, 5). Right turn (North) for 3 km: (0, 5+3) = (0, 8). Point Y is at (0, 8).
Determining displacement from Y to X
Point Y is at (0, 8) and Point X is at (0, 0). To move from Y(0, 8) back to X(0, 0), one must travel 8 units in the negative y-direction, which corresponds to South.
A: 8 km West is wrong because there is no horizontal displacement between Y and X. C: 8 km North is wrong because that would move Vikram further away from the origin. D: 5 km West is incorrect as it does not align with the vertical path back to X.
B is correct because the coordinate calculation shows Vikram is at (0,8) relative to X(0,0), necessitating an 8 km move due South to return to the start.
Always draw a quick X-Y axis sketch for complex turns; treating turns as vector changes makes complex multi-turn problems foolproof.