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.

 

[blue_raver22]

The most standard definition of parallel vectors in vector spaces is that they have the same direction or are multiples of each other. This means that if two vectors, A and B, are parallel, then A = kB for some scalar k. This definition applies to all vector spaces, not just Euclidean spaces. This is because the concept of direction and scalar multiplication applies to any vector space, not just those in Euclidean space.

The definition of parallel in terms of cross product only applies to three-dimensional vectors in Euclidean space. In higher dimensions or in non-Euclidean spaces, the cross product may not exist or may not be a meaningful concept.

It is important to note that the zero vector is parallel to all other vectors in a vector space. This is because any vector multiplied by zero is equal to the zero vector, making it a multiple of the zero vector. Therefore, the zero vector has the same direction as any other vector in the space.

In summary, the formal definition of parallel vectors in vector spaces is that they have the same direction or are multiples of each other. This definition applies to all vector spaces, and the concept of parallel can only be extended to higher dimensions or non-Euclidean spaces with this definition.