Let L1 be the line passing through the points Q1=(−2, 1, 1) and Q2=(−4, 4, 0) and let L2 be the line passing through the point P1=(−6, −6, 0) with direction vector →d=[9, 6, 3]T. Determine whether L1 and L2 intersect. If so, find the point of intersection Q. If not, find a value for the z-coordinate of P1 so the resulting lines do intersect.
qwer
To determine whether the lines L1 and L2 intersect, we can find the point of intersection Q by setting up and solving a system of linear equations.
First, let's find the parametric equations of each line:
For L1:
Direction vector →v1 = Q2 - Q1 = (-4, 4, 0) - (-2, 1, 1) = (-2, 3, -1)
Let t1 be the parameter for L1. The parametric equations of L1 are:
x = -2 - 2t1
y = 1 + 3t1
z = 1 - t1
For L2:
Let P2 = (x, y, z) be any point on L2. Since the direction vector →d = (9, 6, 3), we can write the parametric equations of L2 as:
x = -6 + 9t2
y = -6 + 6t2
z = 0 + 3t2
Now, we'll set up a system of equations to find the values of t1 and t2 where the lines intersect:
-2 - 2t1 = -6 + 9t2 ---(1)
1 + 3t1 = -6 + 6t2 ---(2)
1 - t1 = 3t2 ---(3)
To solve this system of equations, we'll eliminate variables one by one:
From (2), we have: 3t1 - 6t2 = -7 (multiply by 3)
From (3), we have: -t1 - 3t2 = -1 (multiply by -1)
Adding these two equations together:
2t1 - 9t2 = 8 ---(4)
From (1), we have: -2 - 2t1 = -6 + 9t2
Rearranging, we get: 2t1 - 9t2 = -4 ---(5)
Comparing equations (4) and (5), we see that they are the same equation. Therefore, the system is consistent, and the lines intersect.
To find the point of intersection Q, we can substitute the value of t1 or t2 back into the parametric equation of either line. For simplicity, let's use t1.
Using equation (2), we have: 1 + 3t1 = -6 + 6t2
Substituting the value of t1 = -1 into the parametric equations of L2, we get:
x = -6 + 9(-1) = -15
y = -6 + 6(-1) = -12
z = 0 + 3(-1) = -3
Therefore, the point of intersection Q is Q = (-15, -12, -3).
If the lines did not intersect, we can find a value for the z-coordinate of P1 so that the resulting lines do intersect. This is done by solving the system of equations and finding a solution. However, since we found that the lines do intersect, we don't need to find a value for the z-coordinate of P1.