To determine the direction of the man's resultant vector, we can break down his path into its vector components and then find the resultant vector.
1. Initial Vector (Westward Walk)
- The man first walks 500 feet directly west.
- This can be represented as a vector with components:
- [tex]\( x_1 = 500 \)[/tex] feet (west)
- [tex]\( y_1 = 0 \)[/tex] feet
2. Second Vector (South-West Walk)
- The man then turns and walks 371 feet at an angle of [tex]\( 52^\circ \)[/tex] south of west.
- We can decompose this vector into its [tex]\( x \)[/tex] (westward) and [tex]\( y \)[/tex] (southward) components using trigonometry.
- [tex]\( x_2 = 371 \cos 52^\circ \)[/tex]
- [tex]\( y_2 = 371 \sin 52^\circ \)[/tex]
3. Vector Components Calculation
- Calculating the x-component of the second vector:
[tex]\[
x_2 = 371 \cos 52^\circ \approx 371 \times 0.6157 \approx 228.41 \text{ feet west}
\][/tex]
- Calculating the y-component of the second vector:
[tex]\[
y_2 = 371 \sin 52^\circ \approx 371 \times 0.7880 \approx 292.35 \text{ feet south}
\][/tex]
4. Resultant Vector Components
- The total westward component (taking west as negative in traditional Cartesian coordinates):
[tex]\[
x_R = 500 - 228.41 = 271.59 \text{ feet west}
\][/tex]
- The total southward component:
[tex]\[
y_R = 292.35 \text{ feet south}
\][/tex]
5. Direction Calculation
- To find the direction of the resultant vector, we use the arctangent function to determine the direction angle [tex]\(\theta\)[/tex]:
[tex]\[
\theta = \tan^{-2} \left( \frac{292.35}{271.59} \right)
\][/tex]
- Calculating the angle:
[tex]\[
\theta = \tan^{-1} \left( \frac{292.35}{271.59} \right) \approx 47.11^\circ \text{ south of west}
\][/tex]
So, the direction of the man's resultant vector is approximately:
[tex]\[
\theta \approx 47.11^\circ \text{ south of west}
\][/tex]
