Two vector lines are parallel if their direction vectors are scalar multiples of each other and the lines are not coincident. Consider two lines:

These lines are parallel if b1​=kb2​ for some scalar k, but a1​ does not lie on r2​.

Example

Consider two lines given by:

These lines are parallel because their direction vectors are parallel (one is a scalar multiple of the other), but the point (1,1,0) from the second line does not lie on the first line since

simultaneously implies that λ=0,21​ and −21​.