Let the line l0 be defined by x = 1−t, y = 2+3t, z = 1−2t. Determine whether

each given line intersects l0. If it does, give the intersection point. l1 :x=2+t, y=5−2t, z=1+3t
l2 :x=6+2t, y=1+t, z=5+t

To determine if two lines intersect, we need to check if their equations have a common solution. In other words, we need to find values of 't' where the x, y, and z coordinates of both lines are the same.

Let's start with line l1: x = 2 + t, y = 5 - 2t, z = 1 + 3t.

Now, we will substitute these equations into the parametric equations of line l0: x = 1 - t, y = 2 + 3t, z = 1 - 2t.

To find the intersection between l1 and l0, we need to solve the following system of equations:

1 - t = 2 + t
2 + 3t = 5 - 2t
1 - 2t = 1 + 3t

Simplifying these equations, we get:

2t = 1
5t = 3
-5t = 0

From the third equation, we can see that -5t = 0, which means t = 0. Plugging this value into the second equation, we get 5(0) = 3, which is not true. Therefore, lines l1 and l0 do not intersect.

Now, let's move to line l2: x = 6 + 2t, y = 1 + t, z = 5 + t.

Again, we will substitute these equations into the parametric equations of line l0: x = 1 - t, y = 2 + 3t, z = 1 - 2t.

To find the intersection between l2 and l0, we need to solve the following system of equations:

1 - t = 6 + 2t
2 + 3t = 1 + t
1 - 2t = 5 + t

Simplifying these equations, we get:

3t = -5
2t = -1
-3t = 4

From the first equation, we can see that 3t = -5, which means t = -5/3. Plugging this value into the second and third equations, we find the x, y, and z values at the intersection point.

For line l2 and l0 to intersect, t must have the same value for both lines. In this case, since t = -5/3 satisfies all three equations, we can conclude that lines l2 and l0 intersect at the point with coordinates:

x = 1 - t = 1 - (-5/3) = 8/3
y = 2 + 3t = 2 + 3(-5/3) = -3
z = 1 - 2t = 1 - 2(-5/3) = 11/3

Therefore, the intersection point of lines l2 and l0 is (8/3, -3, 11/3).