Parallelism of Vectors

Question

I know that two lines are parallel if they never intersect each other. The conditions for parallel vectors says that if a and b are two vectors then they are parallel if a=kb for k being a scalar. Now if k=1 then a=b implying a is parallel to b that is "a" can be thought of as being parallel to itself. Can one say that every line is also parallel to itself? I think, no. Please share your knowledge with me.

Answer 1

Yes, of course every line is parallel to itself ( it follows by the definition of parallelism because we want parallelism is a relation of equivalence). Notice that if $a$ and $b$ are two vectors and there is k in $R$ with $b$=k $a$, we say that $a$ and $b$ are proportional. Parallelism has a stronger condition: k must be not zero ( because of the consistency of the definition of parallelism between vectors).