Geometry of Vectors

Question

I know the definition of collinear vectors and the condition for collinearity says "Two vectors $a$ and $b$ are collinear if $a=kb$, $k$ being non-zero scalar" but I am confused if $k=0$ then will not they be collinear? If $k=0$, then $a=0$ but $b$ is any vector, since zero vector has arbitrary direction then any vector $b$ and zero vector can be thought of as lying on parallel lines which implies that $b$ and zero vector are collinear i.e $k=0$ is valid for collinearity. Please help me.

Answer 1

By definition, nul-vector is collinear to all other vectors. So, yes, you could say that $k\ge0$, but i guess there are other reasons behind why we don't do so.

Answer 2

We ask that $k \neq 0$ because we want to avoid trivial, useless solutions.

It's analogous to insisting that 1 is not a prime number.

If we allowed $k=0$ then any two vectors would be linearly dependent.

If any two vectors are linearly dependent, then all vectors would be linearly dependent. (Linear dependency is an equivalence relation.)

The only vector space would be the trivial vector space $\{{\bf 0}\}$.