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A taxi drives 6 miles east, then turns and drives at an angle of [tex]$45^{\circ}$[/tex] south of east for 6 more miles. What is the direction of the taxi's resultant vector?

[tex]\[
\begin{array}{c}
|\overrightarrow{R}|=11.09 \text{ miles} \\
\theta=?^{\circ}
\end{array}
\][/tex]


Sagot :

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.