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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.

In the sum \(\vec{A} + \vec{B} = \vec{C}\), vector \(\vec{A}\) has a magnitude of 13.8 m and is angled 36.7° counterclockwise from the +x direction, and vector \(\vec{C}\) has a magnitude of 13.7 m and is angled 24.7° counterclockwise from the -x direction. What are (a) the magnitude and (b) the angle (relative to +x) of \(\vec{B}\)? State your angle as a positive number.

Learn how to solve a vector addition problem involving vectors \(\vec{A}\) and \(\vec{C}\). This problem uses trigonometry to resolve vectors into components and apply the Pythagorean theorem to find the magnitude and angle of vector \(\vec{B}\). Suitable for college-level students studying physics or engineering.

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