Find vector orthogonal to two lines

[chohocvo]

Homework Statement

L1 pass through the points (-2,36,9) and (-8,44,12)
L2 pass through the points (55,-31,7) and (41,-16,13)
Find a point P on L1 and a point Q on L2 so that the vector $$\vec{}PQ$$ is orthgonal to both lines.

Homework Equations

Dot product/ Cross product

The Attempt at a Solution

Equation for L1: x(t) = -2 -6t; y(t) = 36+8t; z(t) = 9 + 3t
Equation for L2: x(s) = 55 -14s; y(s) = -31 +15s; z(s) = 7 +6s
I stuck from here. I can find the intersect pt if the 2 lines are intersect, but this look intersect but have a space between them.

 

[HallsofIvy]

Yes, two lines in space have a "mutual perpendicular" only if they are skew lines.

Every normal plane to L1 has equation -6x+ 8y+ 3z= C for some C and every normal plane to L2 has equation -14x+ 15y+ 6z= D for some D. A line lying in both normal planes (as a mutual perpendicular would) must satisfy both equations. Solve both equations for, say, z and set them equal. That gives you a single equation in x and y so that you can solve for, say, y as a function of x. That will give parametric equations for the mutual perpendicular line, still with unknowns C and D. Determine C and D so that this new line does intersect both L1 and L2.