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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 .

Respuesta :

Refer to the diagram shown below.

Define
[tex]\hat{i} [/tex] = unit vector in the eastern direction
[tex]\hat{j}[/tex] = unit vector n the northern drection.

Then the displacement vectors are
Vector #1:  [tex]1550(cos27^{o} \hat{i} + sin27^{o} \hat{j} ) \, miles = 1381.1\hat{i}+703.7\hat{j}[/tex]
Vector #2: [tex]a(sin41^{o})\hat{i} - cos41^{o}\hat{j} ) = a(0.6561\hat{i} - 0.9873\hat{j} )[/tex]
Vector #3: [tex]b(-cos27^{o} \hat{i} + sin27^{o} \hat{j} ) = b(-0.891\hat{i} + 0.454 \hat{j} )[/tex]

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

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


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