Determine whether L1 and L2 intersect. If so, find the point of intersection Q.

Question

Problem: Let L1 be the line passing through the points Q1 = (4, 3, 1) and Q2 = (5, 1, −1) and let L2 be the line passing through the point P1 = (−7, 15, −7) with direction vector →d=<−4, 4, −4>. Determine whether L1 and L2 intersect. If so, find the point of intersection Q.

I think I am attempting this problem wrong. I first get L1 by doing Q1 + t(Q2), which gives me <5t + 4, t + 3, -t + 1> and get L2 by doing L2 + →d = <-4d - 7, 4d + 15, -4d -7>

After this, I just solved for t and d, and plugged in the values for L1 and L2, but I get the wrong answer on every variant of this question. Can anyone help guide me through this?

Answer 1

Please check the equation of your second line to find a mistake in the direction vector.

Hopefully that solves your problem.

Answer 2

Your parameters are not correct. you need to define parametric equations such that when it is 0, then you get the initial point. In this case then, your parametric equation will be:

for line L1: where t is the vector Q2-Q1

• x-4=t
• y-3=-2t
• z-1-2t

As for L2:

• x+7=-4d
• y-15=4d
• z+7=-4d

in general the parametric equation of a line is

• x-x(naught)=t*v(sub x)
• y-y(naught)=t*v(sub y)
• z-z(naught)=t*v(sub z)

where x,y,z (naught) are x,y, and z components of the point give. v(sub x,y,z) are the x, y, and z components of the direction vector.