To solve this problem, we will determine the vector $\mathbf{AB}$ using the given points $\mathbf{OA} = (-2, -1, 1)$ and $\mathbf{OB} = (1, 3, -1)$, and then calculate the distance from $A$ to $B$.

Step 1: Determine the vector $\mathbf{AB}$

The vector $\mathbf{AB}$ is calculated as: $\mathbf{AB} = \mathbf{OB} - \mathbf{OA}$ Substituting the given values: $\mathbf{AB} = (1, 3, -1) - (-2, -1, 1)$ $\mathbf{AB} = (1 + 2, 3 + 1, -1 - 1)$ $\mathbf{AB} = (3, 4, -2)$

Step 2: Calculate the distance $|\mathbf{AB}|$

The distance from $A$ to $B$ is the magnitude of the vector $\mathbf{AB}$. The formula for the magnitude of a vector $(x, y, z)$ is: $|\mathbf{AB}| = \sqrt{x^2 + y^2 + z^2}$ Substitute the components of $\mathbf{AB}$: $|\mathbf{AB}| = \sqrt{3^2 + 4^2 + (-2)^2}$ $|\mathbf{AB}| = \sqrt{9 + 16 + 4}$ $|\mathbf{AB}| = \sqrt{29}$

Thus, the vector $\mathbf{AB} = (3, 4, -2)$ and the distance from $A$ to $B$ is $\sqrt{29}$.

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Would you like me to break down any of the steps further? Here are five related questions to explore:

1. How do you calculate the midpoint of the line segment joining two points?
2. What is the significance of the direction cosines for a vector like $\mathbf{AB}$?
3. How would this calculation change in a higher-dimensional space?
4. How do we verify if two points are equidistant from a third point?
5. How can the vector $\mathbf{AB}$ be normalized, and what is its significance?

Tip: Always double-check vector operations by performing subtraction or addition component-wise to avoid sign errors.