Let's analyze each option to determine the correctness of each statement. We have two lines, $L_1$ and $L_2$, represented by a system of linear equations, and each option makes a statement about either their angle, distance, orthogonality, or parallelism.

1. Option a: The angle between $L_1$ and $L_2$ is approximately $1.0001 \, \text{rad}$.

   To find the angle between two lines in three-dimensional space, we would calculate the direction vectors of each line and use the dot product to find the cosine of the angle $\theta$ between them: $\cos \theta = \frac{\vec{d_1} \cdot \vec{d_2}}{\|\vec{d_1}\| \|\vec{d_2}\|}$ where $\vec{d_1}$ and $\vec{d_2}$ are the direction vectors of $L_1$ and $L_2$, respectively.

2. Option b: The distance from point $U = \begin{bmatrix} -7 \\ -3 \\ -7 \end{bmatrix}$ to the line $K : \frac{x - 5}{5} = \frac{y + 7}{-5} = \frac{z + 6}{7}$ is approximately $9.2522$.

   The distance from a point to a line in 3D space can be calculated using the formula: $d = \frac{\|\vec{U} - \vec{P}\| \times \vec{d}}{\|\vec{d}\|}$ where $\vec{U}$ is the given point, $\vec{P}$ is a point on the line $K$, and $\vec{d}$ is the direction vector of $K$.

3. Option c: Lines $L_1$ and $L_2$ are orthogonal.

   For orthogonality, the direction vectors of $L_1$ and $L_2$ should satisfy $\vec{d_1} \cdot \vec{d_2} = 0$.

4. Option d: Lines $L_1$ and $L_2$ are parallel.

   For parallelism, the direction vectors of $L_1$ and $L_2$ should be scalar multiples of each other.

5. Option e: None of the above.

Let me know if you want detailed calculations for each of these options or if you have questions about any specific part.

Relative Questions

1. How do we calculate the angle between two lines in 3D space?
2. What is the formula for the distance from a point to a line in three dimensions?
3. How do we determine if two lines are parallel in vector form?
4. What condition must be satisfied for two lines to be orthogonal?
5. How can we verify if none of the statements about the lines are correct?

Tip

For analyzing lines in 3D, always start by determining their direction vectors, as these are essential for finding angles, distances, and relationships like orthogonality and parallelism.