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.
