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Velocity Vectors: Mastery Test

Select the correct answer.

Vector [tex]\(\mathbf{u}\)[/tex] has a magnitude of 5 units and a direction angle of 75°, and vector [tex]\(\mathbf{v}\)[/tex] has a magnitude of 6 units and a direction angle of 210°. What is the component form of the vector [tex]\(\mathbf{u} + \mathbf{v}\)[/tex]?

A. [tex]\(\langle -3.95, 1.83 \rangle\)[/tex]

B. [tex]\(\langle -3.94, 1.89 \rangle\)[/tex]

C. [tex]\(\langle -3.90, 1.89 \rangle\)[/tex]

D. [tex]\(\langle -3.90, 1.83 \rangle\)[/tex]



Answer :

GinnyAnswer
To find the component form of the vector [tex]\( \mathbf{u + v} \)[/tex], we need to follow these steps.

1. Understand the vector magnitudes and directions:
- Vector [tex]\( \mathbf{u} \)[/tex] has a magnitude of 5 units and a direction angle of 75°.
- Vector [tex]\( \mathbf{v} \)[/tex] has a magnitude of 6 units and a direction angle of 210°.

2. Convert the direction angles from degrees to radians:
- The direction angle of [tex]\( \mathbf{u} \)[/tex] in radians: [tex]\( 75° \times \frac{\pi}{180} \)[/tex]
- The direction angle of [tex]\( \mathbf{v} \)[/tex] in radians: [tex]\( 210° \times \frac{\pi}{180} \)[/tex]

3. Calculate the components (x and y) of each vector:
- For vector [tex]\( \mathbf{u} \)[/tex]:
[tex]\[ u_x = \text{magnitude}_u \times \cos(\text{direction angle of } u \text{ in radians}) \][/tex]
[tex]\[ u_y = \text{magnitude}_u \times \sin(\text{direction angle of } u \text{ in radians}) \][/tex]
- For vector [tex]\( \mathbf{v} \)[/tex]:
[tex]\[ v_x = \text{magnitude}_v \times \cos(\text{direction angle of } v \text{ in radians}) \][/tex]
[tex]\[ v_y = \text{magnitude}_v \times \sin(\text{direction angle of } v \text{ in radians}) \][/tex]

4. Sum the components to get the resultant vector [tex]\( \mathbf{r = u + v} \)[/tex]:
- Calculate [tex]\( r_x \)[/tex] and [tex]\( r_y \)[/tex]:
[tex]\[ r_x = u_x + v_x \][/tex]
[tex]\[ r_y = u_y + v_y \][/tex]

From following these steps, we get the resulting component form as:
[tex]\[ \mathbf{r = u + v} = (-3.90, 1.83) \][/tex]

Thus, the correct answer is:
[tex]\[ \textbf{D. } \langle -3.90, 1.83 \rangle \][/tex]

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