https://file.aiquickdraw.com/mathbot/file/1726809177015m7kewwf0.jpg

Let's analyze the problem presented in the image.

You are given two points in 3D space:
- Point **a** = \( \begin{pmatrix} 2 \\ -5 \\ -2 \end{pmatrix} \)
- Point **b** = \( \begin{pmatrix} -2 \\ -8 \\ 4 \end{pmatrix} \)

The question asks for the direction vector **d** of the line passing through these two points.

### Steps to find the direction vector **d**:
The direction vector **d** for a line passing through two points **a** and **b** is calculated by subtracting the coordinates of **a** from **b**:

\[
\mathbf{d} = \mathbf{b} - \mathbf{a}
\]

So, let's compute this:

\[
\mathbf{d} = \begin{pmatrix} -2 \\ -8 \\ 4 \end{pmatrix} - \begin{pmatrix} 2 \\ -5 \\ -2 \end{pmatrix}
\]

Performing the subtraction component-wise:

\[
\mathbf{d} = \begin{pmatrix} -2 - 2 \\ -8 - (-5) \\ 4 - (-2) \end{pmatrix} = \begin{pmatrix} -4 \\ -3 \\ 6 \end{pmatrix}
\]

### Conclusion:
It looks like the answer provided in the image, **d** = \( \begin{pmatrix} 4 \\ 3 \\ -6 \end{pmatrix} \), is incorrect.

The correct direction vector should be:

\[
\mathbf{d} = \begin{pmatrix} -4 \\ -3 \\ 6 \end{pmatrix}
\]

Would you like a more detailed breakdown or have any questions about this solution?

### 5 Related Questions:
1. How do you normalize a direction vector?
2. What is the equation of the line passing through two points in vector form?
3. Can a direction vector be scaled, and does scaling affect the direction?
4. How would you find the distance between two points in 3D space?
5. What are some practical applications of direction vectors in physics?

### Tip:
Always double-check the subtraction when calculating direction vectors to ensure signs are handled correctly.

Find a direction vector, d, for the line passing through the points a = (2, -5, -2) and b = (-2, -8, 4).

Learn how to calculate the direction vector for a line passing through two points in 3D space. This problem uses vector subtraction and applies the formula d = b - a to find the correct direction vector. Suitable for high school students in Grades 10-12.

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