Question

A spelunker is surveying a cave. She follows a passage 120m straight west, then 270m in a direction 45❝ east of south, and then 280 m at 30❝ east of north. After a fourth unmeasured displacement, she finds herself back where she started.

Answer 1

Let the magnitude of A?  be 180m. Let the magnitude of B?  be 210m and it's direction be 45 degrees east of south. Let C?  have the magnitude 280m and a direction of 30 degrees east of north.

0=A? +B? +C? +D?

D? =?(A? +B? +C? )

Dx=?(?180+210sin45+280sin30)=?108.5m

Dy=?(0?210cos45+280cos30)=?94m

|D|=(?108.5)2+(?94)2?????????????????=144m

D?=arctanDy/Dx=arctan?94/?108.5=41deg

Looking over the answers for the angle of a result vector that sits in quadrant 3 I get two possible answers according to the book (the result of the arctanRy/Rx=41):

?=180+40.9 and ?=??180

The way I approached the problem was to recognize that (in the case of this problem) the x and y component of the resultant were in quadrant 3 and then I just drew an angle 41 degrees to the south of the -x axis. My math corresponded to the ? interpretation. However, I don't understand why there are two different definitions of the resultant angle. I understand they're both basically the same (theta sets the angle in Q1 and rotates it whereas phi is simply the angle) but I can't figure out why you need mention both then.