From the image, the problem is about vectors, and it involves finding certain characteristics of two lines represented in vector form. Here's a summary of the given problem:

Given:

The vector equations of two lines $l_1$ and $l_2$ are:

$l_1: \mathbf{r} = 2\mathbf{i} + 3\mathbf{j} + \lambda ( \mathbf{i} - 5\mathbf{j} )$

$l_2: \mathbf{r} = 5\mathbf{i} + 3\mathbf{j} + \mu ( 2\mathbf{i} + \mathbf{j} )$

Where $\lambda$ and $\mu$ are parameters.

To Find:

1. The position vector of the point of intersection of $l_1$ and $l_2$ (if they intersect).
2. The cosine and angle between $l_1$ and $l_2$.

Would you like me to solve this step-by-step, starting with one part at a time? Let me know!

Here are some related questions to expand your understanding:

1. What is the general condition for two vector lines to intersect in 3D space?
2. How do you calculate the angle between two vectors?
3. What role do direction vectors play in the intersection of two lines?
4. How can the dot product and cross product of vectors be used in such problems?
5. What are parallel and skew lines, and how do they relate to intersection problems?

Tip:

Always double-check the direction vectors for both lines, as they play a key role in determining if the lines are parallel, intersecting, or skew.