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.