Determine if line l1 intersects line l3, defined by

l1
[x,y,z] = [4,-3,2] + t[1,8,-3]

l2
[x,y,z] = [1,0,3] + v[4,-5,-9]

To determine if line l1 intersects line l2, we need to check if they have a point in common. We can find the point of intersection by setting the parametric equations of the lines equal to each other and solving for the values of t and v.

Let's write the equations for l1 and l2:

l1: [x, y, z] = [4, -3, 2] + t[1, 8, -3]
l2: [x, y, z] = [1, 0, 3] + v[4, -5, -9]

To find the intersection point, we equate the x, y, and z components of the two lines:

x (l1) = x (l2) ---> 4 + t = 1 + 4v (Equation 1)
y (l1) = y (l2) ---> -3 + 8t = -5v (Equation 2)
z (l1) = z (l2) ---> 2 - 3t = 3 - 9v (Equation 3)

We now have a system of three equations with two unknowns (t and v). We can solve this system to find the values of t and v.