In two lines lie in different plane, can they be parallel to each other?
I am thinking if two lines are parallel to each other, then their direction
cosines must be same so two lines lying in different plane cannot be parallel. Am I missing something? Ni
Example 1. In $R^3$ let $P=\{(x,y,z):y=0\}$ and $Q=\{(x,y,z):z=0\}.$ Let $L_1=\{(x,y,z):y=0\land z=1\}$ and $L_2=\{(x,y,z):z=0\land y=1\}.$ Then $L_1\in P$ and $L_2\in Q .$ And $L_1, l_2$ are parallel.
Example 2. Take a greeting-card, partly opened. Find line-segments on the front and on the back that are parallel.