Definition of Parallel Vectors in Vector Spaces

[Dmobb Jr.]

I was discussing with a friend of mine whether the zero vector is parallel to all other vectors. We came to the conclusion that it probably is but we do not know the formal definition of parallel.

What is the formal definition of parallel? Wolfram says that two vectors are parallel iff their cross product is zero. This, however, is meaningless if the vectors are of dimension greater than 3.

While not the main point of the question, it would also be cool to know if there is some version of parallel for any vector space not just euclidean.

Note: Since there are probably multiple definitions, give the one you think is most standard.

 

[fzero]

Two vectors are parallel if they are scalar multiples of each other. So given two vectors $\mathbf{a},\mathbf{b}$, we can say that $\mathbf{a}$ is parallel to $\mathbf{b}$ if there is a scalar $c$ such that $\mathbf{a} = c \mathbf{b}$.

It is typical to require that $c\neq 0$, but it's not strictly necessary. If we allow $c=0$ then we have the peculiar situation that the zero vector $\mathbf{0}$ is parallel to all other vectors, but no nonzero vector is parallel to $\mathbf{0}$. This would probably be very confusing to people first learning about vectors, so it makes sense to just require that $c\neq 0$ to avoid it.

 

[Dmobb Jr.]

This is what I had initially thought of as the definition but immediately threw it away because it was not commutative. I guess I should never assume such things.

Also I am still curious if anyone knows a definition that works in any vector space.

 

[fzero]

This is what I had initially thought of as the definition but immediately threw it away because it was not commutative. I guess I should never assume such things.

Also I am still curious if anyone knows a definition that works in any vector space.

You can formulate a perfectly fine definitionin two parts:

1. The zero vector is parallel to all other vectors.

2. Nonzero vectors are parallel if they are scalar multiples of each other.

Any version of the definition from this or my earlier post works for any vector space.