To solve this, let's break the problem into its parts.

Part A: What is the angle between $\mathbf{A}$ and $\mathbf{B}$?

We are given that:

• $\mathbf{A} \cdot \mathbf{B} = 28.0 \, \text{m}^2$
• The magnitude of $\mathbf{B}$ is $|\mathbf{B}| = 5.00 \, \text{m}$
• The angle of $\mathbf{B}$ from the positive x-axis is $42.0^\circ$

The dot product formula is:

$\mathbf{A} \cdot \mathbf{B} = |\mathbf{A}| |\mathbf{B}| \cos \theta$

Where $\theta$ is the angle between $\mathbf{A}$ and $\mathbf{B}$, and $|\mathbf{A}|$ is the magnitude of vector $\mathbf{A}$. We know:

$28.0 = |\mathbf{A}| (5.00) \cos \theta$

Simplifying:

$\cos \theta = \frac{28.0}{5.00 |\mathbf{A}|}$

Thus, we need to first determine $|\mathbf{A}|$ to calculate $\theta$.

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Part B: What is the magnitude and direction of $\mathbf{A}$?

We are given that the dot product $\mathbf{B} \cdot \mathbf{C} = 32.0 \, \text{m}^2$ and that vector $\mathbf{C}$ has the same magnitude as $\mathbf{A}$ and its direction is $\delta = 20^\circ$ more than that of $\mathbf{A}$.

Using the same dot product formula for $\mathbf{B} \cdot \mathbf{C}$:

$\mathbf{B} \cdot \mathbf{C} = |\mathbf{B}| |\mathbf{C}| \cos (\theta + 20^\circ)$

Substitute the known values:

$32.0 = 5.00 |\mathbf{A}| \cos (\theta + 20^\circ)$

Now we have two equations:

1. $\cos \theta = \frac{28.0}{5.00 |\mathbf{A}|}$
2. $32.0 = 5.00 |\mathbf{A}| \cos (\theta + 20^\circ)$

We can solve these two equations to find $|\mathbf{A}|$ and $\theta$.

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Solving the system:

Let's solve the two equations step by step to find $|\mathbf{A}|$ and $\theta$.It seems that the solution process took longer than expected. Let's break down the problem in a simpler manner using logical steps rather than solving both equations simultaneously.

Step 1: Calculate the angle $\theta$ between vectors $\mathbf{A}$ and $\mathbf{B}$.

From the first equation:

$\cos \theta = \frac{28.0}{5.00 |\mathbf{A}|}$

Step 2: Substitute $|\mathbf{A}|$ into the second equation.

From the second dot product equation:

$32.0 = 5.00 |\mathbf{A}| \cos (\theta + 20^\circ)$

This leads to a system of two equations that can be solved iteratively by simplifying each expression. If you'd like, I can try solving it with more manual steps or assist with different methods.

Let me know if you want more detailed steps or assistance with a specific part of the process!