To determine the direction of the person's resultant vector after jogging 780 meters south and then 360 meters west, we can use the given information and follow these steps:
1. Draw a Vector Diagram:
Consider the southward jog as a vector pointing downwards (positive y-direction) and the westward jog as a vector pointing to the left (positive x-direction).
2. Identify the Magnitudes:
- Southward distance ([tex]\( d_{\text{south}} \)[/tex]): 780 meters
- Westward distance ([tex]\( d_{\text{west}} \)[/tex]): 360 meters
3. Resultant Vector:
The resultant vector ([tex]\( \vec{R} \)[/tex]) can be found by combining these vectors. It can be considered as the hypotenuse of a right-angled triangle with legs 780 meters and 360 meters.
4. Calculate the Direction:
The direction of the resultant vector relative to the south direction is given by the angle [tex]\( \theta \)[/tex].
- [tex]\( \theta \)[/tex] can be found using the tangent function [tex]\( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \)[/tex].
Since the southward direction is the adjacent side of the triangle and the westward direction is the opposite side, the angle [tex]\( \theta \)[/tex] is given by:
[tex]\[
\theta = \tan^{-1}\left(\frac{\text{westward distance}}{\text{southward distance}}\right)
\][/tex]
Here, [tex]\(\theta = \tan^{-1}\left(\frac{360}{780}\right)\)[/tex].
5. Convert to Degrees:
Many trigonometric calculations provide results in radians, so it's common to convert the result from radians to degrees. However, we'll directly give the angle in degrees here to match the given answer: [tex]\( \theta \approx 24.78^\circ \)[/tex].
Therefore, the direction of the person's resultant vector is approximately [tex]\( 24.78^\circ \)[/tex] west of south.
