Let's break down the solution and understand where the values are derived from for this problem on finding the angle between two lines.

Problem Summary

Given two lines, $L_1$ and $L_2$, we need to find the smallest angle between them. This can be done by calculating the angle between their direction vectors, as shown in the solution.

Step-by-Step Explanation

1. Identifying Direction Vectors:

   • For $L_1$, the line equation is given in parametric form: $L_1: \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix} + t \begin{bmatrix} -1 \\ 1 \\ 2 \end{bmatrix}$ Here, the direction vector $\vec{d_1}$ is extracted as: $\vec{d_1} = \begin{bmatrix} -1 \\ 1 \\ 2 \end{bmatrix}$
   • For $L_2$, the line equation is also in parametric form: $L_2: \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 0 \\ 3 \\ -1 \end{bmatrix} + s \begin{bmatrix} 2 \\ -1 \\ -1 \end{bmatrix}$ Here, the direction vector $\vec{d_2}$ is: $\vec{d_2} = \begin{bmatrix} 2 \\ -1 \\ -1 \end{bmatrix}$

2. Using the Dot Product to Find the Angle:

   • The formula for the angle $\theta$ between two vectors $\vec{d_1}$ and $\vec{d_2}$ is: $\cos \theta = \frac{\vec{d_1} \cdot \vec{d_2}}{|\vec{d_1}| |\vec{d_2}|}$
   • Dot Product $\vec{d_1} \cdot \vec{d_2}$: $\vec{d_1} \cdot \vec{d_2} = (-1)(2) + (1)(-1) + (2)(-1) = -2 - 1 - 2 = -5$

3. Calculating Magnitudes:

   • Magnitude of $\vec{d_1}$: $|\vec{d_1}| = \sqrt{(-1)^2 + 1^2 + 2^2} = \sqrt{1 + 1 + 4} = \sqrt{6}$
   • Magnitude of $\vec{d_2}$: $|\vec{d_2}| = \sqrt{(2)^2 + (-1)^2 + (-1)^2} = \sqrt{4 + 1 + 1} = \sqrt{6}$

4. Calculating $\cos \theta$:

   • Substituting into the formula: $\cos \theta = \frac{-5}{\sqrt{6} \cdot \sqrt{6}} = \frac{-5}{6}$
   • Solving for $\theta$ using the inverse cosine: $\theta = \cos^{-1}\left(-\frac{5}{6}\right)$
   • This results in $\theta \approx 120^\circ$.

5. Smallest Angle Between the Lines:

   • Since angles between lines can have two values that are supplementary, the smallest angle is $180^\circ - 120^\circ = 60^\circ$.

Summary

The derived values for the dot product and magnitudes are based on basic vector operations applied to the direction vectors of each line. The angle calculation uses trigonometric relationships and the concept of supplementary angles to identify the smallest possible angle.

Do you have any further questions, or would you like additional details on any specific part of this solution?

Here are five follow-up questions you might explore:

1. How do we interpret the direction vector in the context of a 3D line?
2. What are some common mistakes to avoid when calculating angles between vectors?
3. How would this approach change if the lines were parallel?
4. Why is it important to find the smallest angle between two lines?
5. What alternative methods could be used to find angles between lines?

Tip: When calculating angles between vectors, ensure you check whether the vectors are parallel or perpendicular, as this can simplify calculations.