Given the coordinates:

• Point $A = (8, 0, 8)$
• Point $B = (0, 4, 0)$
• Point $P = (-6, 10, 6)$

The vector $\mathbf{F}$ goes from point $A$ to point $B$, so:

$\mathbf{F} = \overrightarrow{AB} = B - A$

Step-by-Step Solution:

1. Calculate Vector $\overrightarrow{AB}$:

$\overrightarrow{AB} = (x_B - x_A, y_B - y_A, z_B - z_A) = (0 - 8, 4 - 0, 0 - 8)$

$\overrightarrow{AB} = (-8, 4, -8)$

2. Magnitude of Vector $\overrightarrow{AB}$:

$|\overrightarrow{AB}| = \sqrt{(-8)^2 + 4^2 + (-8)^2}$

$|\overrightarrow{AB}| = \sqrt{64 + 16 + 64} = \sqrt{144} = 12$

3. Unit Vector in the Direction of $\overrightarrow{AB}$:

$\mathbf{u}_{AB} = \frac{\overrightarrow{AB}}{|\overrightarrow{AB}|} = \left( \frac{-8}{12}, \frac{4}{12}, \frac{-8}{12} \right)$

$\mathbf{u}_{AB} = \left( -\frac{2}{3}, \frac{1}{3}, -\frac{2}{3} \right)$

4. Vector from Point $A$ to Point $P$:

$\overrightarrow{AP} = P - A = (-6 - 8, 10 - 0, 6 - 8)$

$\overrightarrow{AP} = (-14, 10, -2)$

5. Projection of Vector $\overrightarrow{AP}$ onto $\overrightarrow{AB}$:

The projection formula is:

$\text{Proj}_{\overrightarrow{AB}} \, \overrightarrow{AP} = (\overrightarrow{AP} \cdot \mathbf{u}_{AB}) \mathbf{u}_{AB}$

First, compute the dot product:

$\overrightarrow{AP} \cdot \mathbf{u}_{AB} = (-14) \left( -\frac{2}{3} \right) + 10 \left( \frac{1}{3} \right) + (-2) \left( -\frac{2}{3} \right)$

$\overrightarrow{AP} \cdot \mathbf{u}_{AB} = \frac{28}{3} + \frac{10}{3} + \frac{4}{3} = \frac{42}{3} = 14$

6. Final Projection Value:

$\text{Proj}_{\overrightarrow{AB}} \, \overrightarrow{AP} = 14 \cdot \left( -\frac{2}{3}, \frac{1}{3}, -\frac{2}{3} \right)$

$\text{Proj}_{\overrightarrow{AB}} \, \overrightarrow{AP} = \left( -\frac{28}{3}, \frac{14}{3}, -\frac{28}{3} \right)$

Conclusion:

The projection of vector $\overrightarrow{AP}$ onto the direction of $\overrightarrow{AB}$ is:

$\left( -\frac{28}{3}, \frac{14}{3}, -\frac{28}{3} \right)$

Would you like further details on this solution, or have any questions?

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Here are some related questions for you:

1. How is the dot product used to find projections?
2. What is the geometric interpretation of a vector projection?
3. Can you calculate the angle between vectors $\overrightarrow{AB}$ and $\overrightarrow{AP}$?
4. How does the magnitude of a vector affect its unit vector?
5. What are some practical applications of vector projections in physics?

Tip: When working with vectors, always visualize them in 3D space to understand their relative directions and magnitudes better!