A parade travels 6 blocks south and then 2 blocks east. What is the direction of the parade's resultant vector?

[tex]\[
\begin{aligned}
|\vec{R}| & = 6.32 \text{ blocks} \\
\theta & = [?]^{\circ}
\end{aligned}
\][/tex]

Round your answer to the nearest hundredth.

[tex]\(\boxed{\text{Enter}}\)[/tex]



Answer :

To find the direction of the parade's resultant vector, let's break down the process step by step.

1. Identify the components of the vector:
- The parade moves 6 blocks south.
- The parade moves 2 blocks east.

These two movements can be represented as the legs of a right triangle where the southward movement (6 blocks) is one leg and the eastward movement (2 blocks) is the other leg.

2. Calculate the magnitude of the resultant vector:
Using the Pythagorean theorem, the magnitude [tex]\( |\vec{R}| \)[/tex] of the resultant vector can be found as:
[tex]\[ |\vec{R}| = \sqrt{(6)^2 + (2)^2} \][/tex]
Simplifying this:
[tex]\[ |\vec{R}| = \sqrt{36 + 4} = \sqrt{40} \approx 6.32 \text{ blocks} \][/tex]

3. Determine the direction (angle [tex]\(\theta\)[/tex]) of the resultant vector:
- The angle [tex]\(\theta\)[/tex] is measured from the east direction counterclockwise to the resultant vector.
- To find [tex]\(\theta\)[/tex], we use the tangent function:
[tex]\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{6}{2} = 3 \][/tex]

4. Calculate the angle [tex]\(\theta\)[/tex]:
Using the arctangent function, we get:
[tex]\[ \theta = \arctan(3) \][/tex]

5. Convert the angle from radians to degrees:
The value in radians can be converted to degrees using a calculator or a conversion factor. The approximate value of [tex]\(\theta\)[/tex] in degrees is:
[tex]\[ \theta \approx 71.57^{\circ} \][/tex]

So, the direction of the parade's resultant vector is approximately [tex]\( 71.57^{\circ} \)[/tex] from the east direction.