The route followed by a hiker consists of three displacement vectors A with arrow, B with arrow, and C with arrow. Vector A with arrow is along a measured trail and is 1550 m in a direction 26.0° north of east. Vector B with arrow is not along a measured trail, but the hiker uses a compass and knows that the direction is 41.0° east of south. Similarly, the direction of vector vector C is 37.0° north of west. The hiker ends up back where she started, so the resultant displacement is zero, or A with arrow + B with arrow + C with arrow = 0. Find the magnitudes of vector B with arrow and vector C with arrow.

Knowing that the sum of the internal angles of a triangle is 180°, we can obtain the internal angles of the triangle formed by the three displacement vectors (see the attached figure, the calculated angles are in red and the given angles in black).

The angles were calculated as follows (see figure):

angle BC = 180°- 90° - 41° - 37 ° = 12°

angle AB = 180° - 90° - 26° +41° = 105°

angle AC = 180° - 105° - 12 = 63°

Once we obtain the internal angles, we can use the sine rule:

sin a/ A = sin b/ B = sin c/ C where "A" is the side opposite to the angle "a", "B" is the side opposite to the angle "b" and "C" is the side the opposite to the angle "c".

Then:

sin 12° / A = sin 63°/ B = sin (105°) / C

sin 12° / 1550 m = sin 63° / B

B = sin 63° * (1550 m / sin 12°) = 6643 m

sin 12° /1550 m = sin 105° /C

C = sin 105° * (1550 m / sin 12°) = 7201 m