The route followed by a hiker consists of three displacement vectors , , and . vector is along a measured trail and is 1550 m in a direction 27.0° north of east. vector 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 is 29.0° north of west. the hiker ends up back where she started, so the resultant displacement is zero, or + + = 0. find the magnitudes of vector and vector .

Refer to the diagram shown below.

Define
 = unit vector in the eastern direction
 = unit vector n the northern drection.

Then the displacement vectors are
Vector #1:  
Vector #2: 
Vector #3: 

Because the vector sum of all three vector is zero, therefore
1381.1 +0.6561a - 0.891b = 0
703.7 - 0.9873a + 0.454b = 0

That is,
  0.6561a - 0.891b  = -1381.1         (1)
-0.9873a + 0.454b = -703.7        (2)

From (1), obtain
b = 0.7364a + 1550.1                 (3)
Substitute (3) into (2).
-0.9873a + 0.454(0.7364a + 1550.1) = - 703.7
-0.653a = -1407.4
a = 2155.3 mles
From (3), obtain
b = 3137.2 mles

The magnitudes of the two displacement vectors are
2155.3 miles and 3137.2 miles.