A plane leaves the airport in Galisteo and flies 170 km at 68 degrees east of no

Question

A plane leaves the airport in Galisteo and flies 170 km at 68 degrees east of north, then changes direction to fly 230 km at 48 degrees south of east, and then changes direction to fly 120 km west, after which it makes an immediate emergency landing in a pasture. When the airport sends out a rescue crew, in which direction and how far should this crew fly to go directly to the plane?

I am not sure if I drew my diagram angles correctly to get the correct sines and cosines.

I drew on xy coordinates with 68 degress between y-axis and 170km vector, then 230km vector from tip of other, with 48 degree angle at the tail on the outside, pointing in negative y direction south of east, lastly the 120 km vector again at the tip of the last one pointing due west, -120 km.

I came up with <170sin68, 170cos68>, <230cos48, -230 sin48>, <-120, 0> = <191.52, -107.24>. Magnitude = 219.50 or 220 km. Direction tan^-1 (-107.24/191.52)= -29 degrees.

Please check for any problems with any of my work. Also, can the degrees be answered as a negative (if my answer is correct) or does it need to be a positive angle?

THANKS!

Explanation / Answer

r   = 170sin68i + 170cos68j + 230cos48i - 230sin48j - 120 i   = 191.52i   - 107.24

|r| = 219.50 km

now we come to direction.

since, x-component is positive and y-component is negative.

so we will consider east-south quadrant.

angle = tan-1(107.24/191.52) = 29.25 degrees below east in east-south

so direction is 29.25 degrees south to east.

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