Determine whether or not the parametric equations of two lines intersect. L 1 :

Question

Determine whether or not the parametric equations of two lines intersect.

L1:
x = 5 + 3s
y = -3 - 2s
z = 4 + s

and
L2:
x = 3-t
y = -4 + 3t
z = 5 - 2t

The answer is yes at P(2, -1, 3).

I just need to know how to do this type of problem.

Explanation / Answer

If L1 and L2 intersect, there will be a point (x,y,z) where both lines cross with each other. Therefore, x, y, and z will be the same at that point. if x=x, y=y and z=z for both lines, there will be values for s and t that will satisfy the following equations; x=x y=y z=z (from both lines) then: 5+3s = 3-t .........(1) -3-2s = -4 +3t ...........(2) 4 + s = 5- 2t ...........(3) from (3) s = 5 -2t -4 (solving for x) s = 1 - 2t ...........(4) (4) in (1) 5 + 3 (1-2t) = 3- t 5 +3 -6t = 3-t 8 -6t = 3-t 8 - 3 = 6t -t 5 = 5t t = 1 -----> substituting this value in (4) ---> s = -1 t = 1 ans s = -1 satisfy (1) and (3)... in order to be an intersection should satisfy also (2) t= 1 and s= -1 in (2) -3 -2 (-1) = -4 +3(1) -3 +2 = -4 + 3 -1 = -1 .....TRUE it satisfy (2) replacing either t =1 in L1 or s=-1 in L 2 will give us the same answer ----> the intersection point x = 3 -(1) = 2 y = -4 +3(1) = -1 z = 5 - 2(1) = 3 L1 and L2 will intersect at the point (x,y,z) = (2,-1,3)

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