To solve for the maximum value of the magnitude of $\mathbf{B} - \mathbf{A}$, where $\mathbf{B}$ and $\mathbf{A}$ are vectors, we will consider the following approach:

Step-by-step process:

1. Define the vectors: Let $\mathbf{A}$ and $\mathbf{B}$ be two vectors. The vector $\mathbf{B} - \mathbf{A}$ represents the vector that points from $\mathbf{A}$ to $\mathbf{B}$.

2. Magnitude of a vector: The magnitude (or length) of a vector $\mathbf{V} = \mathbf{B} - \mathbf{A}$ is given by: $|\mathbf{B} - \mathbf{A}| = \sqrt{(B_x - A_x)^2 + (B_y - A_y)^2 + (B_z - A_z)^2}$ where $B_x, B_y, B_z$ and $A_x, A_y, A_z$ are the components of $\mathbf{B}$ and $\mathbf{A}$, respectively.

3. Maximizing the magnitude: The magnitude is maximized when the vectors $\mathbf{B}$ and $\mathbf{A}$ are pointing in opposite directions. In that case, the angle $\theta$ between them is 180°, and the distance between the points represented by the vectors is greatest.

Thus, the maximum value of $|\mathbf{B} - \mathbf{A}|$ is equal to the sum of the magnitudes of $\mathbf{A}$ and $\mathbf{B}$: $|\mathbf{B} - \mathbf{A}|_{\text{max}} = |\mathbf{B}| + |\mathbf{A}|$

Final Answer:

The maximum value of $|\mathbf{B} - \mathbf{A}|$ is $|\mathbf{A}| + |\mathbf{B}|$.

Would you like more details or further clarification on this?

---

Here are 5 related questions for further understanding:

1. How do you calculate the magnitude of a vector in a 2D coordinate system?
2. What is the geometric interpretation of subtracting one vector from another?
3. How does the angle between two vectors affect the magnitude of their difference?
4. Can the magnitude of $\mathbf{B} - \mathbf{A}$ ever be negative? Why or why not?
5. How would you find the maximum value of $|\mathbf{B} - \mathbf{A}|$ if the vectors were given in polar coordinates?

Tip: The dot product between two vectors can help determine the angle between them, which is useful when calculating vector differences.