https://file.aiquickdraw.com/mathbot/file/172668915988090tk9e5n.jpg

Let's break down the information from the image to solve the problem:

1. **Point A coordinates**: \( A(3, 1, 5) \).
2. **Equation of line \( l \)**: 
   \[
   \mathbf{r} = \begin{pmatrix} 8 \\ 0 \\ -1 \end{pmatrix} + t \begin{pmatrix} -6 \\ 1 \\ 4 \end{pmatrix}
   \]
   Here, the direction vector of the line \( l \) is \( \mathbf{d}_l = \begin{pmatrix} -6 \\ 1 \\ 4 \end{pmatrix} \).
3. **Task**: Find the coordinates of point \( B(x, y, z) \) such that vector \( \overrightarrow{AB} \) is perpendicular to line \( l \). This means \( \overrightarrow{AB} \cdot \mathbf{d}_l = 0 \), where \( \mathbf{d}_l \) is the direction vector of line \( l \).

### Step-by-Step Solution:

Let point \( B \) have coordinates \( (x, y, z) \). Then, the vector \( \overrightarrow{AB} \) from point \( A(3, 1, 5) \) to point \( B(x, y, z) \) is:
\[
\overrightarrow{AB} = \begin{pmatrix} x - 3 \\ y - 1 \\ z - 5 \end{pmatrix}
\]
For \( \overrightarrow{AB} \) to be perpendicular to the line \( l \), the dot product of \( \overrightarrow{AB} \) with the direction vector of line \( l \), \( \mathbf{d}_l = \begin{pmatrix} -6 \\ 1 \\ 4 \end{pmatrix} \), must be zero:
\[
\overrightarrow{AB} \cdot \mathbf{d}_l = (x - 3)(-6) + (y - 1)(1) + (z - 5)(4) = 0
\]
Expanding this equation:
\[
-6(x - 3) + (y - 1) + 4(z - 5) = 0
\]
Simplifying:
\[
-6x + 18 + y - 1 + 4z - 20 = 0
\]
\[
-6x + y + 4z - 3 = 0
\]
This is the equation that point \( B(x, y, z) \) must satisfy.

Would you like me to solve this equation further or proceed with specific values for \( x, y, z \)? Let me know if you'd like further clarifications! Here are a few related questions:

### Follow-up Questions:
1. What are the parametric coordinates of any point on line \( l \)?
2. Can we solve this equation for \( y \) or \( z \) given a specific \( x \)-value?
3. How can we verify if the vector \( \overrightarrow{AB} \) is indeed perpendicular to \( l \)?
4. How do we compute the distance between point \( A \) and any point on line \( l \)?
5. What if we were to calculate point \( B \) assuming it's on line \( l \)?

### Tip:
When working with perpendicular vectors, remember that their dot product will always be zero, as this is the definition of perpendicularity.

The point A has coordinates (3, 1, 5) and the line l has equation r = (8, 0, -1) + t(-6, 1, 4). Find the coordinates of the point B on l such that AB is perpendicular to l.

This problem involves finding the coordinates of a point B such that the vector AB is perpendicular to a line l. The problem uses key concepts from vector algebra, including dot products and line equations. Suitable for advanced high school students in Grades 11-12.

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