A treasure map directs you to start at a palm tree and walk due north for 10.0 m

Question

A treasure map directs you to start at a palm tree and walk due north for 10.0 m. You are then to turn 90° and walk 13.0 m; then turn 90° again and walk 5.00 m. For each of the four possible locations of the treasure, give the distance from the palm tree and the direction, expressed as an angle measured counterclockwise from north. List the possible locations in counterclockwise (CCW) order, starting with the one which is closest to due east.
____m and _____° (location closest to due east)
____ m and _____° (next location in the CCW direction)
____ m and _____° (next location in the CCW direction)
____ m and _____° (next location in the CCW direction)

Explanation / Answer

In the CCW direction, starting closest to due East, We have 4 possible treasure locations, each with a position vector, r_n for n = 1,2,3,4 After walking North 10.0 meters, r_1: turn right, turn right r_2: turn right, turn left r_3: turn left, turn right r_4: turn left, turn left --- r_1 = (13.0,-5.00)m r_1 = [(13.0m)^2 + (-5.00m)^2]^1/2 = 13.9 m direction (angle from East): theta_1 = tan^-1[r_y/r_x] = tan^-1[-5.00m/13.0m] = tan^-1[-0.385] = 21.0 deg South of East --- By symmetry arguments, all displacements are thus: r_1 = 13.9 meters at 21.0 deg South of East r_2 = 13.9 m at 21.0 deg North of East r_3 = 13.9 m at 21.0 deg North of West r_4 = 13.9 m at 21.0 deg South of West

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