Skew lines in three dimensions

Question

My question is : how can i prove if a line r in 3D In parametric form is skew with the z-axis Knowing that the line r for example has equation : $$ \left\{ \begin{array}{c} X=1+t \\ Y=2-t\\ Z=1+t\end{array} \right. $$ I tried to prove that the cross product of the direction vector of r and z-axis is different from zero (not parallel) and the dot product also different then zero ( not perpendicular) but what about the intersection between them ?

Answer 1

The line intersects the z-axis if $X=Y=0$. But that is impossible in your case.

To check that the line is not parallel to the z-axis, just notice that its direction vector $(1,-1,1)$ is not a multiple of $(0,0,1)$.

Answer 2

The lines are parallel if their direction vector are the multiple of each other (or the same), meanwhile if it's not it can be either intersecting or skew. They are skew if you cant find the intersection point of those two line