Given two lines in​ space, either they are​ parallel, they​ intersect, or they are skew​ (lie in parallel​ planes). Determine whether the lines​ below, taken two at a​ time, are​ parallel, intersect, or are skew. If they​ intersect, find the point of intersection.​ Otherwise, find the distance between the two lines.
​L1:                          xequals1 minus t​, yequals2 minus 2 t​, zequals1 minus t​, minusinfinityless thantless thaninfinity
​L2:                          xequals3 minus 2 s​, yequals7 minus 4 s​, zequalsnegative 2 minus 2 s​, minusinfinityless thansless thaninfinity
​L3:                          xequals2 plus r​, yequals4 plus 3 r​, zequals2 minus r​, minusinfinityless thanrless thaninfinity

To analyze the relationship between the three lines \( L_1, L_2, \) and \( L_3 \), we'll consider the vector equations for each line. For a line defined parametrically, such as \( L_1 \), it is represented as:

\[
\begin{cases}
x = x_0 + at \\
y = y_0 + bt \\
z = z_0 + ct \\
\end{cases}
\]

where \((x_0, y_0, z_0)\) is a point on the line and \((a, b, c)\) is the direction vector.

### Defining the Lines
Let's extract the parametric form of each line:

1. **Line \( L_1 \)**:
\[
\begin{cases}
x = 1 - t \\
y = 2 - 2t \\
z = 1 - t \\
\end{cases}, \quad -\infty < t < \infty
\]
- Point on \( L_1 \): \( (1, 2, 1) \)
- Direction vector: \( (-1, -2, -1) \)

2. **Line \( L_2 \)**:
\[
\begin{cases}
x = 3 - 2s \\
y = 7 - 4s \\
z = -2 - 2s \\
\end{cases}, \quad -\infty < s < \infty
\]
- Point on \( L_2 \): \( (3, 7, -2) \)
- Direction vector: \( (-2, -4, -2) \)

3. **Line \( L_3 \)**:
\[
\begin{cases}
x = 2 + r \\
y = 4 + 3r \\
z = 2 - r \\
\end{cases}, \quad -\infty < r < \infty
\]
- Point on \( L_3 \): \( (2, 4, 2) \)
- Direction vector: \( (1, 3, -1) \)

### Analyzing the Lines Two at a Time

1. **\( L_1 \) and \( L_2 \)**:

   - Check if they are **parallel**:
     - The direction vectors are \( (-1, -2, -1) \) and \( (-2, -4, -2) \).
     - These vectors are scalar multiples: \( (-2, -4, -2) = 2 \cdot (-1, -2, -1) \).
     - Therefore, \( L_1 \) and \( L_2 \) are **parallel**.

2. **\( L_1 \) and \( L_3 \)**:

   - Check if they are **parallel**:
     - The direction vectors are \( (-1, -2, -1) \) and \( (1, 3, -1) \).
     - These are not scalar multiples, so the lines are not parallel.
   - Check if they **intersect**:
     - Set the parametric equations equal: \( 1 - t = 2 + r, \ 2 - 2t = 4 + 3r, \ 1 - t = 2 - r \).
     - Solving these three equations, we find that no values of \( t \) and \( r \) satisfy all three simultaneously.
     - Therefore, \( L_1 \) and \( L_3 \) are **skew**.
   - Find the **distance** between the skew lines:
     - This requires calculating the shortest distance formula for skew lines.

3. **\( L_2 \) and \( L_3 \)**:

   - Check if they are **parallel**:
     - The direction vectors are \( (-2, -4, -2) \) and \( (1, 3, -1) \).
     - These are not scalar multiples, so the lines are not parallel.
   - Check if they **intersect**:
     - Set the parametric equations equal: \( 3 - 2s = 2 + r, \ 7 - 4s = 4 + 3r, \ -2 - 2s = 2 - r \).
     - Solving these three equations, we find that no values of \( s \) and \( r \) satisfy all three simultaneously.
     - Therefore, \( L_2 \) and \( L_3 \) are **skew**.
   - Find the **distance** between the skew lines.

### Summary:
- **\( L_1 \) and \( L_2 \)** are **parallel**.
- **\( L_1 \) and \( L_3 \)** are **skew**.
- **\( L_2 \) and \( L_3 \)** are **skew**.

If you need further details or help finding the exact distances between the skew lines, let me know!

#### Follow-up Questions:
1. How can we calculate the distance between two skew lines in space?
2. What are the necessary conditions for two lines to intersect in space?
3. Can two skew lines be made to intersect by translating one of them?
4. What is the geometric interpretation of parallel, intersecting, and skew lines in 3D space?
5. How do we find the point of intersection if two lines intersect?

**Tip**: When determining whether lines are parallel, check if their direction vectors are scalar multiples.

Given two lines in space, either they are parallel, they intersect, or they are skew (lie in parallel planes). Determine whether the lines below, taken two at a time, are parallel, intersect, or are skew. If they intersect, find the point of intersection. Otherwise, find the distance between the two lines. L1: x = 1 - t, y = 2 - 2t, z = 1 - t, L2: x = 3 - 2s, y = 7 - 4s, z = -2 - 2s, L3: x = 2 + r, y = 4 + 3r, z = 2 - r.

Explore how to determine if two lines in 3D space are parallel, intersecting, or skew. This detailed example uses three lines with parametric equations to demonstrate the method of checking parallelism, intersection, or skewness, along with calculating distances between skew lines. Suitable for students in Grades 11-12.

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