To solve the problem of determining the direction of the taxi's resultant vector, let's follow the steps systematically:
1. Initial Information:
- The taxi drives 6 miles east.
- Then, it turns and drives at an angle of [tex]\(45^\circ\)[/tex] south of east for another 6 miles.
2. Conversions and Components:
- The 45-degree angle south of east needs to be broken down into its eastward and southward components.
3. Calculating Components of the Second Segment:
- The eastward distance (component) can be found using [tex]\( \cos(45^\circ) \)[/tex]:
[tex]\[
\text{East Component} = 6 \times \cos(45^\circ) = 6 \times \frac{1}{\sqrt{2}} \approx 4.2426 \text{ miles}
\][/tex]
- The southward distance (component) can be found using [tex]\( \sin(45^\circ) \)[/tex]:
[tex]\[
\text{South Component} = 6 \times \sin(45^\circ) = 6 \times \frac{1}{\sqrt{2}} \approx 4.2426 \text{ miles}
\][/tex]
4. Summing the Components:
- The total eastward distance is the sum of the initial 6 miles and the eastward component of the second part:
[tex]\[
\text{Total East Component} = 6 + 4.2426 \approx 10.2426 \text{ miles}
\][/tex]
- The total southward distance is simply the southward component of the second part:
[tex]\[
\text{Total South Component} = 4.2426 \text{ miles}
\][/tex]
5. Determining the Direction of the Resultant Vector:
- To find the direction, use the tangent function to find the angle [tex]\( \theta \)[/tex] south of east:
[tex]\[
\tan(\theta) = \frac{\text{Total South Component}}{\text{Total East Component}} = \frac{4.2426}{10.2426}
\][/tex]
- Calculate [tex]\( \theta \)[/tex] by taking the arctangent (inverse tangent):
[tex]\[
\theta = \tan^{-1} \left( \frac{4.2426}{10.2426} \right) \approx 0.3927 \text{ radians}
\][/tex]
- Convert this angle from radians to degrees:
[tex]\[
\theta \text{ (in degrees)} = 0.3927 \times \frac{180}{\pi} \approx 22.5^\circ
\][/tex]
Therefore, the direction of the taxi's resultant vector is approximately [tex]\(22.5^\circ\)[/tex] south of east.
