To find the point of intersection for the two parametric lines, we need to solve for the values of $t$ and $u$ where the two vectors are equal, i.e., the position vectors of the two lines coincide.

The parametric equations for the lines are given as:

1. $\mathbf{v}(t) = \langle -1, -6, -9 \rangle t + \langle 1, 3, -2 \rangle$
2. $\mathbf{w}(u) = \langle -5, 4, -5 \rangle u + \langle -17, -3, -44 \rangle$

Step 1: Write out the parametric equations for each coordinate (x, y, z).

• For the first line $\mathbf{v}(t)$, the coordinates are: $x_1 = -1t + 1, \quad y_1 = -6t + 3, \quad z_1 = -9t - 2$

• For the second line $\mathbf{w}(u)$, the coordinates are: $x_2 = -5u - 17, \quad y_2 = 4u - 3, \quad z_2 = -5u - 44$

Step 2: Set the coordinates equal to each other.

For the point of intersection, the coordinates of $\mathbf{v}(t)$ and $\mathbf{w}(u)$ must be equal. Therefore, we have the following system of equations:

$-1t + 1 = -5u - 17 \quad \text{(x-component)}$ $-6t + 3 = 4u - 3 \quad \text{(y-component)}$ $-9t - 2 = -5u - 44 \quad \text{(z-component)}$

Step 3: Solve the system of equations.

1. From the first equation: $-1t + 1 = -5u - 17$ Simplify: $-t + 1 = -5u - 17 \quad \Rightarrow \quad -t = -5u - 18 \quad \Rightarrow \quad t = 5u + 18$

2. Substitute $t = 5u + 18$ into the second equation: $-6t + 3 = 4u - 3$ Substitute $t = 5u + 18$: $-6(5u + 18) + 3 = 4u - 3$ Simplify: $-30u - 108 + 3 = 4u - 3 \quad \Rightarrow \quad -30u - 105 = 4u - 3$ Combine like terms: $-30u - 105 = 4u - 3 \quad \Rightarrow \quad -30u - 4u = 105 - 3 \quad \Rightarrow \quad -34u = 102$ Solve for $u$: $u = \frac{102}{-34} = -3$

3. Now substitute $u = -3$ into $t = 5u + 18$: $t = 5(-3) + 18 = -15 + 18 = 3$

Step 4: Find the point of intersection.

Now, substitute $t = 3$ into the parametric equation for $\mathbf{v}(t)$ to find the coordinates of the point of intersection: $\mathbf{v}(3) = \langle -1, -6, -9 \rangle(3) + \langle 1, 3, -2 \rangle = \langle -3, -18, -27 \rangle + \langle 1, 3, -2 \rangle = \langle -2, -15, -29 \rangle$

Thus, the point of intersection is $\boxed{(-2, -15, -29)}$.

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Would you like further clarification or help with another problem?

Here are 5 related questions:

1. How do you interpret the parametric equations of a line in 3D space?
2. What is the geometric meaning of the intersection of two parametric lines in 3D?
3. How do you solve systems of equations involving more than two variables?
4. What are the steps for checking if two lines in 3D space are parallel, skew, or intersecting?
5. Can you find the intersection of two parametric lines using matrices?

Tip: When solving systems of equations, substitution and elimination methods can often simplify the process.