https://file.aiquickdraw.com/mathbot/file/1728567740363l5i8weqa.jpg

Let's break down the problem step by step, focusing on vectors \(\vec{A}\) and \(\vec{B}\) from the figure.

### Given:
- Vector \(\vec{A}\) has a magnitude of 44.0 at an angle of 28.0° above the x-axis.
- Vector \(\vec{B}\) has a magnitude of 26.5 at an angle of 56.0° below the negative y-axis (equivalent to \(180^\circ + 56^\circ = 236^\circ\) counterclockwise from the positive x-axis).

### Part A:
You are asked to find \(\vec{B} - \vec{A}\), which involves subtracting the components of these vectors.

#### Step 1: Find the components of \(\vec{A}\)
The vector \(\vec{A}\) can be split into x and y components:
- \(A_x = A \cdot \cos(\theta_A)\)
- \(A_y = A \cdot \sin(\theta_A)\)

Where:
- \(A = 44.0\)
- \(\theta_A = 28.0^\circ\)

So:
\[
A_x = 44.0 \cdot \cos(28.0^\circ) \approx 38.9
\]
\[
A_y = 44.0 \cdot \sin(28.0^\circ) \approx 20.7
\]

#### Step 2: Find the components of \(\vec{B}\)
Similarly, \(\vec{B}\) can be split into x and y components:
- \(B_x = B \cdot \cos(\theta_B)\)
- \(B_y = B \cdot \sin(\theta_B)\)

Where:
- \(B = 26.5\)
- \(\theta_B = 236.0^\circ\)

So:
\[
B_x = 26.5 \cdot \cos(236^\circ) \approx -14.7
\]
\[
B_y = 26.5 \cdot \sin(236^\circ) \approx -21.9
\]

#### Step 3: Calculate \(\vec{B} - \vec{A}\)
Now subtract the components:
- \((B - A)_x = B_x - A_x = -14.7 - 38.9 \approx -53.6\)
- \((B - A)_y = B_y - A_y = -21.9 - 20.7 \approx -42.6\)

Thus:
\[
\vec{B} - \vec{A} = (-53.6, -42.6)
\]

### Part B:
For this part, you need to calculate \(\vec{A} - \vec{B}\), which is simply the reverse of Part A.

#### Step 1: Swap the subtraction:
- \((A - B)_x = A_x - B_x = 38.9 - (-14.7) = 38.9 + 14.7 = 53.6\)
- \((A - B)_y = A_y - B_y = 20.7 - (-21.9) = 20.7 + 21.9 = 42.6\)

Thus:
\[
\vec{A} - \vec{B} = (53.6, 42.6)
\]

---

Would you like further clarifications or more details on the solution?

Here are 5 related questions:
1. How do we find the magnitude of a vector from its components?
2. What is the geometric interpretation of vector subtraction?
3. How would the answer change if the angles for \(\vec{A}\) and \(\vec{B}\) were negative?
4. How can we verify the correctness of the calculated vector components?
5. What other methods can be used to perform vector subtraction?

**Tip:** Always double-check the angles, ensuring you measure from the positive x-axis counterclockwise for standard direction, unless otherwise specified.

Given the vectors A and B shown in the figure, determine B - A and A - B by finding their components.

Learn how to subtract two vectors, B and A, using their components derived from trigonometric functions. This problem focuses on breaking down vector magnitudes and directions into x and y components using sine and cosine functions.

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