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?