The vector $\mathbf{v}$ is given by its initial point $(-2, 5)$ and terminal point $(-2, 1)$. To find the magnitude (or norm) $\|\mathbf{v}\|$ of the vector, we first calculate the vector's components by subtracting the initial point from the terminal point.

$\mathbf{v} = \left( -2 - (-2), 1 - 5 \right) = \left( 0, -4 \right)$

Now, the magnitude of the vector $\mathbf{v} = (0, -4)$ is given by the formula:

$\|\mathbf{v}\| = \sqrt{(0)^2 + (-4)^2} = \sqrt{0 + 16} = \sqrt{16} = 4$

Thus, the exact value of $\|\mathbf{v}\|$ is 4.

Would you like further details or have any questions?

Here are 5 related questions:

1. What is the formula for finding the magnitude of a vector in 3 dimensions?
2. How do you calculate the direction of a vector?
3. How would the magnitude of the vector change if both components were negative?
4. What does the magnitude of a vector represent geometrically?
5. Can a vector have a negative magnitude?

Tip: The magnitude of a vector is always a non-negative value since it's the length of the vector in space.