Can skew lines share a normal vector?
I know that if I take the cross product of two vectors then I will be able to find the normal vector to the plane created by $a\times b$.
And if I take the cross product of the direction vectors from the skew lines and it is non zero, then I will get the normal vector to a plane created by the direction vectors.
And from there I could come up with an equation for the plane created by those two vectors?
How would I do that?
Best Answer
If you have a pair of skew lines with direction vectors ${\bf a}$ and ${\bf b}$, then since they are skew, their direction vectors are not parallel. Non-parallel vectors will always yield a nonzero cross product. So ${\bf n} = {\bf a} \times {\bf b}$ will (for skew lines) always be a nonzero vector.
So just as with any nonzero vector, you can use ${\bf n}$ as a normal for a plane. But to determine a plane you need a normal vector and a point.
If you choose a point on one of the skew lines, you'll get a plane containing that line and the other skew line will be parallel to your plane.
If you choose a point not on either of the skew lines, you'll get a plane which is parallel to both of the lines but contains neither line.