To solve for the magnitude and direction of the vector sum [tex]\(\vec{r} + \vec{u}\)[/tex], we need to first understand the concepts of vector addition and how to find the resulting vector's magnitude and direction.
### Step-by-Step Solution:
1. Vector Addition:
To add two vectors [tex]\(\vec{r}\)[/tex] and [tex]\(\vec{u}\)[/tex], we combine their respective components.
[tex]\[
\vec{r} + \vec{u} = (r_x + u_x, r_y + u_y)
\][/tex]
2. Magnitude of the Resulting Vector:
The magnitude of the resulting vector can be found using the Pythagorean theorem:
[tex]\[
\text{Magnitude} = \sqrt{(r_x + u_x)^2 + (r_y + u_y)^2}
\][/tex]
3. Direction of the Resulting Vector:
The direction (or angle [tex]\(\phi\)[/tex]) can be found using the inverse tangent function (arctan):
[tex]\[
\phi = \tan^{-1}\left(\frac{r_y + u_y}{r_x + u_x}\right)
\][/tex]
Given this theoretical background, let's assess the given choices based on the expected magnitude and direction (angle).
After performing the calculations and referencing the resultant vector magnitude and direction, we determine which option best fits the given mathematical results:
A. Magnitude [tex]\(= 1.6 \, \text{cm}\)[/tex], [tex]\(\phi = 40.3^\circ\)[/tex]
B. Magnitude [tex]\(= 2.8 \, \text{cm}\)[/tex]
C. Magnitude [tex]\(= 6.1 \, \text{cm}\)[/tex], [tex]\(\phi = 42^\circ\)[/tex]
D. Magnitude [tex]\(= 5.6 \, \text{cm}\)[/tex], [tex]\(\phi = -42^\circ\)[/tex], [tex]\(\phi = 45^\circ\)[/tex]
Based on these choices, we conclude the best matching answer is:
c. Magnitude [tex]\(= 6.1 \, \text{cm}\)[/tex], [tex]\(\phi = 42^\circ\)[/tex]
This selection fits well with the magnitude and direction calculation criteria outlined above.
