Displacement of a vector

Question

The summit of a mountain, 2072 m above base camp, is measured on a map to be 4576 m horizontally from the camp in a direction 32.4 degrees west of north. Choose the x axis east, y axis north and z axis up. What are the components of the displacement vector from camp to summit, what is the magnitude?

The teacher totally didnt explain this and the problem makes no sense to me HELP!

Answer 1

the z component is easy, z = 2072 = 2072 m up

draw a vector in the x-y plane with length 4576 at an angle 32.4 deg anti-clockwise from the positive y axis

you can see that the x component is

x = -4576sin32.4 ≈ -2452 = 2452 m west

and

y = 4576cos32.4 ≈ 3864 = 3864 m north

so the displacement vector is

(x, y, z) ≈ (-2452, 3864, 2072) m

its magnitude, using pythagoras is

√[x² + y² + z²] = √[(-2452)² + (3864)² + (2072)²] ≈ 5023 m

or more simply

√[(4576)² + (2072)²] ≈ 5023 m

.,.,.,