https://file.aiquickdraw.com/mathbot/file/1721973526523ra7mptbw.png

Given the points and values:

- \( A \equiv (2, a, 4) \)
- \( B \equiv (6, b, 4) \)
- \( C \equiv (6, c, 0) \)
- \( D \equiv (2, d, 0) \)
- \( M \equiv (4, e, 2) \)

With the given values:
- \( a = 2 \)
- \( b = 3 \)
- \( c = 7 \)
- \( d = 2 \)
- \( e = 8 \)

### Finding the Vector \(\overrightarrow{AB}\)

Calculate the coordinates of points \( A \) and \( B \):
- \( A = (2, 2, 4) \)
- \( B = (6, 3, 4) \)

The vector \(\overrightarrow{AB}\) is given by:
\[ \overrightarrow{AB} = B - A = (6 - 2, 3 - 2, 4 - 4) = (4, 1, 0) \]

### Equation of the Line \( l_1 \)

The given line \( l_1 \) has the equation:
\[ \mathbf{r}_1 = 6\mathbf{i} + 4\mathbf{j} + \lambda (\mathbf{i} + \mathbf{j}) \]
where \(\lambda\) is a scalar parameter.

Rewriting the equation:
\[ \mathbf{r}_1 = (6 + \lambda) \mathbf{i} + (4 + \lambda) \mathbf{j} \]

The direction vector of \( l_1 \) is:
\[ \mathbf{d}_1 = (\mathbf{i} + \mathbf{j}) = (1, 1, 0) \]

### Parallelism of \(\overrightarrow{AB}\) and \( l_1 \)

To show \(\overrightarrow{AB}\) is parallel to \( l_1 \), their direction vectors must be proportional. 

The direction vector of \( \overrightarrow{AB} \) is \( (4, 1, 0) \).

Check if there exists a constant \( k \) such that:
\[ (4, 1, 0) = k (1, 1, 0) \]

Clearly, \( k = 4 \) works as:
\[ (4, 1, 0) = 4 (1, 1, 0) \]

Therefore, \(\overrightarrow{AB}\) is parallel to \( l_1 \).

### Line \( l_2 \)

The line \( l_2 \) passes through points \( C \) and \( M \).

Calculate the coordinates of points \( C \) and \( M \):
- \( C = (6, 7, 0) \)
- \( M = (4, 8, 2) \)

Find the direction vector \(\overrightarrow{CM}\):
\[ \overrightarrow{CM} = M - C = (4 - 6, 8 - 7, 2 - 0) = (-2, 1, 2) \]

Equation of line \( l_2 \):
\[ \mathbf{r}_2 = (6, 7, 0) + \mu (-2, 1, 2) \]

### Summary

- \(\overrightarrow{AB} = (4, 1, 0)\)
- \( l_1 \) has direction vector \( (1, 1, 0) \)
- \( \overrightarrow{AB} \) is parallel to \( l_1 \) as \( (4, 1, 0) = 4 (1, 1, 0) \)

Thus, \(\overrightarrow{AB}\) is indeed parallel to \( l_1 \).

Would you like more details or have any questions?

Here are some related questions you might find interesting:
1. How to find the equation of a line in 3D space?
2. What is the significance of direction vectors in geometry?
3. How can we prove two vectors are parallel?
4. How to find the distance between a point and a line in space?
5. What are some applications of vector algebra in physics?

**Tip:** When dealing with vector problems, always clearly define your vectors and check for proportional relationships to determine parallelism or orthogonality.

Explore how vectors and line parallelism are applied in 3D geometry. Solve a problem involving vector calculations and line equations, demonstrating the parallelism between a vector and a line in 3D space. Suitable for advanced high school students and beyond.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation