Let L1 be the line passing through the points Q1=(−2, −5, 4) and Q2=(4, −1, 2) and let L2 be the line passing through the point P1=(10, 8, −7) with direction vector →d=[1, −1, 2]T. Determine whether L1 and L2 intersect. If so, find the point of intersection Q.

I did the same question about a week ago,

all you have to do is change a few numbers.

Without doing the actual steps again, watch if near the end,
LS = RS , then they do intersect, and you can find the point
or
LS ≠ RS, then they would NOT intersect

To determine whether the lines L1 and L2 intersect, we need to check if there is a common point between them.

Step 1: Find the equations of the lines L1 and L2.

For line L1:
The direction vector →v1 of L1 can be found by subtracting the coordinates of Q1 and Q2.
→v1 = Q2 - Q1 = (4, -1, 2) - (-2, -5, 4) = (6, 4, -2)

The parametric equations of L1 can be written as:
x = -2 + 6t
y = -5 + 4t
z = 4 - 2t

For line L2:
The parametric equations of L2 can be written as:
x = 10 + t
y = 8 - t
z = -7 + 2t

Step 2: Set up an equation to find the intersection point.

To find if L1 and L2 intersect, we need to find values for t in the equations of L1 and L2 that satisfy them simultaneously. We can equate the x, y, and z coordinates of both lines to find the values of t.

-2 + 6t = 10 + t (equating the x coordinates)
-5 + 4t = 8 - t (equating the y coordinates)
4 - 2t = -7 + 2t (equating the z coordinates)

Step 3: Solve the system of equations.

6t - t = 10 + 2
4t + t = 8 + 5
4t + 2t = -7 + 4

Simplifying the equations gives us:
5t = 12
5t = 13
6t = -3

None of these equations have a solution, which means the lines L1 and L2 do not intersect.