# Skew lines in three dimensions

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 ?

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 $$(0,0,1)$$.