Q: If two vectors form a plane in 3 space, then how come two vectors can be “skew” and never intersect?

A: Skew vectors that never intersect do not start both at the origin (of course, because if they did, the origin would be the intersection point and they would not be skew!)

Vectors that are skew never intersect – you have to present them in parametric form so that you’re sure the vector doesn’t start at the origin. An example is:

v1 = ( 1, 0, 0 ) + t( 0, 1, 0 ) ;
v2 = ( -1, 0, 0 ) + t( 0, 0, 1 ) ;

These are the two vectors on either side of the origin. One starts at ( 1, 0, 0 ), so right on the +x-axis, and goes STRAIGHT up.

The other vector starts on the -x-axis, at (-1,0,0) and goes STRAIGHT in the direction of +z.

Because one vector goes straight up and down, and the other vector goes back and forth in z, they’ll never hit each other. But the vectors AREN’T parallel.