Let L1 be the line passing through the points Q1=(2, −2, 5) and Q2=(−4, −6, 3) and let L2 be the line passing through the point P1=(7, 4, 0) with direction vector →d=[−1, −2, 3]T. Determine whether L1 and L2 intersect. If so, find the point of intersection Q.
L1:
r1= (2,-2,5) + s<6,4,2>
x = 2 + 6s
y = -2 + 4s
z = 5 + 2s
L2:
r2 = (7,4,0) + t<-1,-2,3>
x = 7 - t
y = 4 - 2t
z = 0 + 3t
If they intersect, then their corresponding x, y , and z values must be the
same for some constant s and t
I will set the y's and z's equal,
-2 + 4s = 4 - 2t ---> 4s + 2t = 6 ----> 2s + t = 3
5 + 2s = 3t
2s - 3t = -5
subtract those last two:
4t = 8
t = 2
sub into 2s+t=3
2s + 2 = 3
s = 1/2
then in L1, z = 5 + 2s = 5+5/2 = 15/2
in L2, z = 3t = 6
I did not get the same value, so the two lines do not intersect
better check my calculations, only had one coffee so far.
To determine whether L1 and L2 intersect, we need to check if there is a point that lies on both lines. If a point exists, we can find the coordinates of that point as the point of intersection Q.
Line L1 can be represented as a parametric equation of the form:
x = 2 + t*(-6)
y = -2 + t*(-4)
z = 5 + t*(3)
Here, t is a parameter that represents different points along the line.
Line L2 can be represented as:
x = 7 + s*(-1)
y = 4 + s*(-2)
z = 0 + s*(3)
Where s is a parameter representing different points along the line.
To find the point of intersection, we need to set the x, y, and z coordinates of both lines equal to each other and solve for the values of t and s:
2 + t*(-6) = 7 + s*(-1)
-2 + t*(-4) = 4 + s*(-2)
5 + t*(3) = 0 + s*(3)
Simplifying the equations:
-6t + s = 5
-4t + 2s = 6
3t - 3s = -5
Now we have a system of linear equations. We can solve it to find the values of t and s. By substituting the first equation into the second equation, we get:
-4t + 2(-6t + s) = 6
-4t - 12t + 2s = 6
-16t + 2s = 6
-8t + s = 3
Now we have a system with two equations:
-16t + 2s = 6
-8t + s = 3
Solving these equations will give us the values of t and s. Once we have these values, we can substitute them back into one of the original parametric equations to find the coordinates of the point of intersection Q.