We are given two vector equations for the lines $\mathbf{r}_1(t)$ and $\mathbf{r}_2(s)$ as follows:

$\mathbf{r}_1(t) = \langle -2, 4, 7 \rangle + t \langle 0, 1, 3 \rangle$ $\mathbf{r}_2(s) = \langle -11, 2, 13 \rangle + s \langle -3, 0, 4 \rangle$

To find the point of intersection $P$ of the two lines, we need to set the vector equations equal to each other because at the point of intersection, both lines share the same position vector. So:

$\mathbf{r}_1(t) = \mathbf{r}_2(s)$

Expanding both equations:

$\mathbf{r}_1(t) = \langle -2, 4, 7 \rangle + t \langle 0, 1, 3 \rangle = \langle -2, 4 + t, 7 + 3t \rangle$ $\mathbf{r}_2(s) = \langle -11, 2, 13 \rangle + s \langle -3, 0, 4 \rangle = \langle -11 - 3s, 2, 13 + 4s \rangle$

Now, equate the corresponding components:

1. $-2 = -11 - 3s$
2. $4 + t = 2$
3. $7 + 3t = 13 + 4s$

Solve these three equations for $t$ and $s$.

Step 1: Solve the second equation for $t$

$4 + t = 2 \implies t = 2 - 4 = -2$

Step 2: Solve the first equation for $s$

$-2 = -11 - 3s \implies -2 + 11 = -3s \implies 9 = -3s \implies s = -3$

Step 3: Check consistency using the third equation

$7 + 3t = 13 + 4s$ Substitute $t = -2$ and $s = -3$ into the equation:

$7 + 3(-2) = 13 + 4(-3) \implies 7 - 6 = 13 - 12 \implies 1 = 1$

Since the third equation is consistent, the solution is valid.

Step 4: Find the point of intersection $P$

Substitute $t = -2$ into $\mathbf{r}_1(t)$ or $s = -3$ into $\mathbf{r}_2(s)$ to find the coordinates of $P$.

Using $\mathbf{r}_1(t)$: $\mathbf{r}_1(-2) = \langle -2, 4 + (-2), 7 + 3(-2) \rangle = \langle -2, 2, 1 \rangle$

Thus, the point of intersection is:

$P = \langle -2, 2, 1 \rangle$

Would you like further details on the steps? Here are some additional related questions:

1. How do you find the parametric equations for a line given two points?
2. Can lines that don't intersect still have the same direction vectors?
3. What if two lines are parallel but do not intersect? How would you handle that case?
4. How can we check if two lines are skew (not intersecting and not parallel)?
5. How would the solution change if the lines were defined in 2D instead of 3D?

Tip: When solving systems of equations with parametric variables, always check for consistency across all components to ensure there is an intersection point.