To determine the direction of the parade's resultant vector after traveling 6 blocks south and then 2 blocks east, we can use trigonometry.
1. Understand the Problem:
The parade travels 6 blocks south and 2 blocks east. We need to find the direction of the resultant vector concerning the south direction, expressed as an angle.
2. Resultant Vector Magnitude:
The magnitude of the resultant vector, [tex]\(|\vec{R}|\)[/tex] is already provided to be 6.32 blocks. This is determined using the Pythagorean theorem:
[tex]\[
|\vec{R}| = \sqrt{(6)^2 + (2)^2} = 6.32 \text{ blocks}
\][/tex]
3. Determine the Direction Angle [tex]\(\theta\)[/tex]:
We need to calculate the angle [tex]\(\theta\)[/tex] of the resultant vector with respect to the southern direction. This can be found using the arctangent function. The angle [tex]\(\theta\)[/tex] is calculated using the ratio of the lengths of the eastward and southward segments of the journey:
[tex]\[
\tan(\theta) = \frac{\text{blocks east}}{\text{blocks south}} = \frac{2}{6}
\][/tex]
[tex]\[
\theta = \arctan\left(\frac{2}{6}\right)
\][/tex]
4. Convert to Degrees:
The arctangent function will give us the angle in degrees since we typically measure angles in degrees.
5. Round to the Nearest Hundredth:
Finally, we round the resulting angle to the nearest hundredth. After the calculation, we find that:
[tex]\[
\theta \approx 18.43^\circ
\][/tex]
Therefore, the direction of the parade's resultant vector is:
[tex]\[
\theta = 18.43^{\circ}
\][/tex]
Hence, the complete solution is:
[tex]\[
|\vec{R}| = 6.32 \text{ blocks}, \quad \theta = 18.43^\circ
\][/tex]
