To find a vector $\mathbf{V}$ that moves points along the line $L$, which passes through the points $(-8, -1)$ and $(2, -12)$, we can use the difference between these two points.

1. Let the two points be: $P_1 = (-8, -1) \quad \text{and} \quad P_2 = (2, -12)$

2. The vector $\mathbf{V}$ that moves points along $L$ can be obtained by subtracting the coordinates of $P_1$ from $P_2$: $\mathbf{V} = P_2 - P_1 = (2 - (-8), -12 - (-1))$

3. Simplifying the components: $\mathbf{V} = (2 + 8, -12 + 1) = (10, -11)$

Therefore, the vector $\mathbf{V} = (10, -11)$ moves points along the line $L$.

Would you like more details or have any questions?

Here are some related questions you might explore:

1. How would you find the slope of line $L$ using points $(-8, -1)$ and $(2, -12)$?
2. What is the equation of line $L$ in slope-intercept form?
3. How can you determine if another point lies on the line $L$ defined by $(-8, -1)$ and $(2, -12)$?
4. How can you use vector $\mathbf{V}$ to write a parametric equation for line $L$?
5. If $L$ were extended indefinitely, how could you represent its direction with a unit vector?

Tip: When finding a vector along a line passing through two points, subtract the coordinates of the initial point from the terminal point to obtain the direction vector.