To find the coordinates of a point that is two-thirds of the way from point $A(3, 4, -1)$ to point $B(3, -1, -4)$, we use the formula for the coordinates of a point dividing a line segment in a given ratio:

$P(x, y, z) = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}, \frac{mz_2 + nz_1}{m+n} \right)$

Here:

• $A(x_1, y_1, z_1) = (3, 4, -1)$
• $B(x_2, y_2, z_2) = (3, -1, -4)$
• The ratio is $2:1$, so $m = 2$ and $n = 1$.

Step 1: Compute the coordinates

1. For $x$: $x = \frac{2x_2 + 1x_1}{2 + 1} = \frac{2(3) + 1(3)}{3} = \frac{6 + 3}{3} = 3$

2. For $y$: $y = \frac{2y_2 + 1y_1}{2 + 1} = \frac{2(-1) + 1(4)}{3} = \frac{-2 + 4}{3} = \frac{2}{3}$

3. For $z$: $z = \frac{2z_2 + 1z_1}{2 + 1} = \frac{2(-4) + 1(-1)}{3} = \frac{-8 - 1}{3} = \frac{-9}{3} = -3$

Step 2: Final Answer

The coordinates of the point two-thirds of the way from $A(3, 4, -1)$ to $B(3, -1, -4)$ are: $P(3, \frac{2}{3}, -3)$

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Do you want further clarification or visualization of this result?

Here are 5 related questions to explore further:

1. What would be the coordinates if the ratio were reversed to $1:2$?
2. How do you determine the midpoint of a line segment in three dimensions?
3. Can we use vectors to solve this problem more efficiently?
4. What is the formula for a general point dividing a segment in $m:n$ ratio in 3D space?
5. How would this approach extend to dividing a segment into multiple equal parts?

Tip: Always verify your result by checking if the point lies between the given endpoints when dividing a segment.