Question

Vector v has a magnitude of 8 and makes an angle of 12° with the horizontal. Vector w has a magnitude of 3 and makes an angle of 71° with the horizontal. What is the magnitude of v + w? A. 6.948 units B. 9.885 units C. 11 units D. 13.302 units

Answer 1

To find the magnitude of the vector sum \( \mathbf{v} + \mathbf{w} \), we can first break down each vector into its components.

### Step 1: Find the components of vector \( \mathbf{v} \)

The components of vector \( \mathbf{v} \) can be calculated as follows:

\[
v_x = v \cos(\theta_v) = 8 \cos(12^\circ)
\]

\[
v_y = v \sin(\theta_v) = 8 \sin(12^\circ)
\]

Calculating \( v_x \) and \( v_y \):

\[
v_x = 8 \cos(12^\circ) \approx 8 \times 0.9781 \approx 7.865
\]
\[
v_y = 8 \sin(12^\circ) \approx 8 \times 0.2079 \approx 1.664
\]

### Step 2: Find the components of vector \( \mathbf{w} \)

Next, we find the components of vector \( \mathbf{w} \):

\[
w_x = w \cos(\theta_w) = 3 \cos(71^\circ)
\]
\[
w_y = w \sin(\theta_w) = 3 \sin(71^\circ)
\]

Calculating \( w_x \) and \( w_y \):

\[
w_x = 3 \cos(71^\circ) \approx 3 \times 0.3256 \approx 0.977
\]
\[
w_y = 3 \sin(71^\circ) \approx 3 \times 0.9460 \approx 2.838
\]

### Step 3: Add the components of \( \mathbf{v} \) and \( \mathbf{w} \)

Now, we can add the corresponding components of \( \mathbf{v} \) and \( \mathbf{w} \):

\[
R_x = v_x + w_x = 7.865 + 0.977 \approx 8.842
\]
\[
R_y = v_y + w_y = 1.664 + 2.838 \approx 4.502
\]

### Step 4: Calculate the magnitude of \( \mathbf{R} = \mathbf{v} + \mathbf{w} \)

The magnitude of the resultant vector \( \mathbf{R} \) can be found using the Pythagorean theorem:

\[
\| R \| = \sqrt{R_x^2 + R_y^2}
\]

Calculating the magnitude:

\[
\| R \| = \sqrt{(8.842)^2 + (4.502)^2} \approx \sqrt{78.311 + 20.258} \approx \sqrt{98.569} \approx 9.93
\]

### Step 5: Final result

After calculating the above, the closest value to our calculated magnitude is:

\[
\| \mathbf{R} \| \approx 9.885 \text{ units}
\]

Thus, the answer is \( \boxed{9.885} \) units.