How do you find the intersection of two 3d lines on a 3d plane?

You need to more precise in your question. There is no guarantee two lines interection in 3-d space. If they do intersect, they must lie in the same plane.

And that plane would only need to be 2-D, aside from the trivial case where the lines are the same If the two lines do intersect then x value for the two lines will be equal, as well as the y value & z value, in the equations of the two lines Using parametric equations, If line1 is given by: x = x1 + a*t, y = y1 + b*t, and z = z1 + c*t; and line 2 is: x = x2 + d*s, y = y2 + e*s, and z = z2 + f*s, where x1,y1,z1 & x2,y2,z2 are starting points a,b,c & d,e,f are direction vectors, and s & t are the parameters Since the x values must be equal, set x1+a*t = x2+d*s, and y1+b*t = y2+e*s, then you have two equations and two unknowns. Solve for t and s, then substitute into the z equations to find the z coordinate (they will both come up with the same z-value if indeed the lines do intersect). Even if both equations use 't' as the parameter, you need to treat them as two independent variables(so change one of them to s), since the lines could be changing differently as the parameter variable changes.

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