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The image contains a mathematical problem related to vectors and geometry in 3D space. Here is a breakdown of the questions from the image:

### Given Points:
- Point A: \( A(0, 3, -1) \)
- Point B: \( B(1, 9, 3) \)
- Point C: \( C(2, 5, -1) \)

#### Questions:
1. **(a)** Verify that the vector \( 4\mathbf{i} + 4\mathbf{j} - 7\mathbf{k} \) is perpendicular to the plane ABC, and write down the Cartesian equation of the plane.
2. **(b)** The line \( L_1 \) passes through point \( A \) and point \( P(6, 5, -8) \). Write down the vector equation of the line \( L_1 \).
3. **(c)** The line \( L_2 \) is the reflection of the line \( L_1 \) in the plane ABC. Find a vector equation of \( L_2 \).

### Let's begin by addressing part (a):

#### (a) Perpendicular Vector and Cartesian Equation of the Plane:
To verify that \( \mathbf{n} = 4\mathbf{i} + 4\mathbf{j} - 7\mathbf{k} \) is perpendicular to the plane, we need to:
- Find a normal vector to the plane ABC.
- Confirm that the given vector is a scalar multiple of this normal vector.
- Then, write the equation of the plane using the general form \( ax + by + cz = d \), where \( \mathbf{n} = (a, b, c) \) is the normal vector.

I'll compute the necessary steps now.### Results for part (a):

1. The normal vector to the plane ABC is \( \mathbf{n}_{ABC} = (-8, 8, -10) \).
2. The given vector \( 4\mathbf{i} + 4\mathbf{j} - 7\mathbf{k} \) is **not** parallel to the normal vector of the plane ABC (i.e., it is not a scalar multiple of the normal vector).
3. The Cartesian equation of the plane ABC is:
   \[
   -8x + 8y - 10z = 34
   \]
   or equivalently:
   \[
   8x - 8y + 10z = -34
   \]

Now, let's proceed to part (b), where we need to find the vector equation of the line \( L_1 \) that passes through points \( A(0, 3, -1) \) and \( P(6, 5, -8) \).### Part (b): Vector Equation of Line \( L_1 \)

The direction vector of line \( L_1 \), which passes through points \( A(0, 3, -1) \) and \( P(6, 5, -8) \), is:
\[
\mathbf{d}_{L_1} = (6, 2, -7)
\]

The vector equation of the line \( L_1 \) is:
\[
\mathbf{r} = \mathbf{A} + t \cdot \mathbf{d}_{L_1} = (0, 3, -1) + t(6, 2, -7)
\]
where \( t \) is a parameter.

Next, for part (c), we need to find the vector equation of the line \( L_2 \), which is the reflection of \( L_1 \) in the plane ABC.

Would you like to continue with part (c) or have any specific clarifications so far?

The points A, B and C are given by coordinates A(0, 3, -1), B(1, 9, 3), and C(2, 5, -1). Verify that the vector 4i + 4j - 7k is perpendicular to the plane ABC and write down the Cartesian equation of the plane. The line L1 passes through A and the point P(6, 5, -8). Write down the vector equation of the line L1. The line L2 is the reflection of the line L1 in the plane ABC. Find a vector equation of L2.

This detailed solution addresses a 3D geometry problem involving verifying vector perpendicularity to a plane, deriving the Cartesian equation, and finding vector equations of lines, including reflection in a plane. Suitable for Grades 11-12.

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