Batman is surveying a cave, which is now known as the bat cave. He follows a passage 210.0m straight east, then 80.00m in a direction 60o west of north, then 150.0m at 45o west of south. After a fourth unmeasured displacement he finds himself back where he started. Find the magnitude and direction of the fourth displacement.

Question

Batman is surveying a cave, which is now known as the bat cave. He follows a passage 210.0m straight east, then 80.00m in a direction 60o west of north, then 150.0m at 45o west of south. After a fourth unmeasured displacement he finds himself back where he started. Find the magnitude and direction of the fourth displacement.

Answer 1

translating all directions to standard trig notation, and using vectors:

210(cos0, sin0) + 80(cos150, sin150) + 150(cos225,sin225) + a(cosθ, sinθ) = (0,0
(210,0) + (-69.282, 40) + (-106.066, -106.066) + a(cosθ,sinθ) = 0
a(cosθ,sinθ) = (-34.652, 66.066)
a(cosθ,sinθ) = 74.602(cos117.677, sin117.677)

form a valid statement from my conclusion