Masters of Vectors are welcomed!?

Question

An explorer is caught in a whiteout (in which the snowfall is so thick that the ground cannot be distinguished from the sky) while
returning to base camp. He was supposed to travel due north for 5.4 km, but when the snow clears, he discovers that he actually traveled 7.8 km at 53° north of due east.

(a) How far must he now travel to reach base camp?
km

(b) In what direction must he travel?
° (counterclockwise from due east)

Answer 1

First, Determine his position.
If we say he started at 0,0
and the camp is at 0,5.4

during the white out he traveled to
7.8*cos(53),7.8*sin(53)

4.69, 6.23

If we now translate the camp to 0,0
his position is
4.69, (6.23-5.4)
or
4.69, .83

the distance is
sqrt(4.69^2+.83^2)
=4.76 km

and the angle is
arctan(.83/4.69) south of west
so he must travel 4.76km 10 degrees south of west to get to camp

j

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Answer 2

From a point draw a line 5.4 km straight up, and another 7.8 km long at an angle of 53 degrees up from horizontal, which is to say 90-53=37 degrees to the right of the first line, and a third line from the tips of the other two lines. Then use trigonometry to solve the triangle for the length of the third line and the values of the other two angles in the triangle. All vector problems are basically triangle problems.

Answer 3

x=7.8 * cos53