The problem you shared asks to find the magnitude and angle of vector B in the vector equation:

$\vec{A} + \vec{B} = \vec{C}$

Given Data:

• Magnitude of $\vec{A}$ = 13.8 m, and its angle = 36.7° (counterclockwise from the +x direction)
• Magnitude of $\vec{C}$ = 13.7 m, and its angle = 24.7° (counterclockwise from the -x direction)

We need to:

1. Find the magnitude of $\vec{B}$.
2. Find the angle of $\vec{B}$.

Solution Approach:

We will break $\vec{A}$ and $\vec{C}$ into their components and use the vector equation to solve for $\vec{B}$.

Step 1: Resolve vectors $\vec{A}$ and $\vec{C}$ into components

• For $\vec{A}$, the angle is given as 36.7° from the +x axis: $A_x = A \cdot \cos(36.7^\circ)$ $A_y = A \cdot \sin(36.7^\circ)$ Substituting $A = 13.8 \, \text{m}$: $A_x = 13.8 \cdot \cos(36.7^\circ)$ $A_y = 13.8 \cdot \sin(36.7^\circ)$

• For $\vec{C}$, the angle is given as 24.7° counterclockwise from the -x axis, so its angle relative to the +x axis is 180° - 24.7° = 155.3°: $C_x = C \cdot \cos(155.3^\circ)$ $C_y = C \cdot \sin(155.3^\circ)$ Substituting $C = 13.7 \, \text{m}$: $C_x = 13.7 \cdot \cos(155.3^\circ)$ $C_y = 13.7 \cdot \sin(155.3^\circ)$

Step 2: Use the vector sum equation

The components of $\vec{B}$ are found using:

$B_x = C_x - A_x$ $B_y = C_y - A_y$

Step 3: Find the magnitude and angle of $\vec{B}$

• Magnitude of $\vec{B}$: $B = \sqrt{B_x^2 + B_y^2}$
• Angle of $\vec{B}$: $\theta_B = \tan^{-1}\left(\frac{B_y}{B_x}\right)$

I'll now calculate the values.The magnitude and angle of vector $\vec{B}$ are as follows:

• Magnitude of $\vec{B}$: $23.65 \, \text{m}$
• Angle of $\vec{B}$: $-173.88^\circ$ (relative to the +x axis)

Since the problem asks for the angle as a positive number, we can express the angle as:

$\theta_B = 360^\circ - 173.88^\circ = 186.12^\circ$

Summary:

• (a) Magnitude of $\vec{B}$: $23.65 \, \text{m}$
• (b) Angle of $\vec{B}$: $186.12^\circ$

Would you like further clarification or details on the steps?

Here are some related questions to expand your understanding:

1. How are vector components related to their magnitudes and angles?
2. Why do we resolve vectors into x and y components to solve vector addition problems?
3. What happens to the angle when a vector points in the negative x-direction?
4. Can you use the Pythagorean theorem for vector addition in all cases?
5. How would this process change if the vectors were in 3D space?

Tip: Always double-check angle references (from +x or -x) when resolving vector directions.