Starting from their base in the national park, a group of

Question

bushwalkers travel 1.5 km at a true bearing of 030°, then 3.5 km
at a true bearing of 160°, and then 6.25 km at a true bearing of
300°. How far, and at what true bearing, should the group walk
to return to its base?

Answer 1

arghh, that "bearing" stuff is like chalk on a blackboard to true
mathematicians. Let's translate to the standard trig-angle notation
bearing 30 ---> 60°
bearing 160 ---> 290°
bearing 300 ---> 150°

so we could just use vectors: If P is your final destination,
vector OP = 1.5(cos60,sin60) + 3.5(cos290,sin290) + 6.25(cos150,sin150)
= (-3.4656, 1.1351) , which is in quadrant II
magnitude = √(-3.4656^2 + 1.1351^2) = 3.65 km
angle in real math notation = 162°

converting 162° back to "bearing" would be 288 degrees
BUT we want to walk from P to O, so the bearing should be
288-180 or 108° (the -180 because your turned yourself around)

3.65 km on a bearing of 108°

check my arithmetic