Perpendicular lines in three dimensions

Question

If a line , say, in the z axis , is said to be perpendicular to the x-y plane , does it imply that any line drawn on the x-y plane will be perpendicular to the given line ?

Answer 1

Yes indeed in parametric form

• $z$ axis is defined by $t(0,0,1)$
• a line in $x-y$ plane has form $(x_0,y_0,0)+t(a,b,0)$

and the dot product of the direction vectors is

$$(0,0,1)\cdot (a,b,0)=0$$

thus they always are orthogonal.

Answer 2

Yes. These are called Skew Lines: https://en.wikipedia.org/wiki/Skew_lines