# Definition of Parallel Vectors in Vector Spaces

• Dmobb Jr.
In summary, the formal definition of parallel for two vectors in any vector space is that the zero vector is parallel to all other vectors, and nonzero vectors are parallel if they are scalar multiples of each other. This definition holds true regardless of the dimension of the vectors or the type of vector space.

#### 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.

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.

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.

Dmobb Jr. said:
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.

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.

## What is the definition of parallel vectors in vector spaces?

Parallel vectors in vector spaces are defined as two or more vectors that have the same or opposite direction, but may have different magnitudes. This means that if the vectors were to be placed end-to-end, they would lie on the same or parallel lines.

## How can you determine if two vectors are parallel in a vector space?

To determine if two vectors are parallel in a vector space, you can use the scalar multiplication rule. If the scalar multiple of one vector is equal to the other vector, then they are parallel. This means that one vector is a multiple of the other, in other words, they have the same direction.

## Can parallel vectors have different lengths?

Yes, parallel vectors can have different lengths as long as they have the same direction. The magnitude or length of a vector does not affect its direction, so two vectors can have different lengths and still be considered parallel.

## Do parallel vectors have to be in the same vector space?

Yes, parallel vectors must be in the same vector space. Vector spaces have specific rules and properties that determine if two vectors are parallel, so it is necessary for the vectors to be in the same vector space for this definition to apply.

## How are parallel vectors used in real life applications?

Parallel vectors have many real-life applications, such as in physics, engineering, and computer graphics. In physics, parallel vectors are used to represent the direction and magnitude of force, while in engineering, they are used in designing structures and calculating forces. In computer graphics, parallel vectors are used to create 3D images and animations.