A spelunker is surveying a cave. She follows a passage 180 m straight west, then 210 m in a direction 45 degrees east of south, and then 280 m at 30 degrees east of north. After a fourth unmeasured displacement she finds herself back where she started. Use a scale drawing to determine the magnitude and direction of the fourth displacement.
Obviously, I refuse to use a scale and try to get a proper solution. Here's my solution
Since she went back to where she started, the magnitude of the displacement vector is zero so its obvious that we should
$0=\sqrt{(-180+210cos(45)+280cos(60)+x)^2 + (210sin(45)+280sin(60)+y)^2}$.
Which tells us that the magnitude of the final vector is 405.74. However, the solutions manual tells me that the magnitude is 144 (after careful measurement).
Can someone explain to me what I'm doing wrong here? Unlike my previous posts asking help for Real Analysis proofs, this is actually school related
The mistake in your approach is that you're not considering the angles correctly. When the spelunker moves in different directions, the angles are measured with respect to the reference axes (east and north).