Zero vector

A null vector is a vector with zero magnitude that originates at the origin. It is represented by the symbol $$ \vec{0} = \overrightarrow{OO} $$

In essence, the null vector is one where the starting point (O) and the endpoint (O) are the same.

The null vector has no direction or orientation, and its magnitude (length) is exactly zero.

Note: The null vector serves as the zero element of a vector space $ V $ and acts as the additive identity for vectors $ \vec{v} $ within that space. $$ \vec{v} + \vec{0} = \vec{v} \:\:\: \forall \ \vec{v} \in V $$ Consequently, adding two null vectors also results in a null vector: $$ \vec{0} + \vec{0} = \vec{0} $$ Similarly, the additive inverse of the null vector is the null vector itself: $$ -\vec{0} = \vec{0} $$

Each vector space has a unique null vector, regardless of its dimension. This vector is defined as having zero magnitude and all components equal to zero.

Its uniqueness is guaranteed by the properties of vector spaces, particularly the definition of the additive identity.

Example

For example, in the vector space $ \mathbb{R} $, which consists of real numbers, the null vector corresponds to the number $ 0 $, the additive identity in this space.

$$ \vec{0} = \begin{pmatrix} 0 \end{pmatrix} $$

In the vector space $ \mathbb{R}^2 $, consisting of ordered pairs of real numbers $ (x, y) $, the null vector is $ \mathbf{0} = (0, 0) $, where both components are $ 0 $.

$$ \vec{0} = \begin{pmatrix} 0 \\ 0 \end{pmatrix} $$

More generally, in a vector space $ \mathbb{R}^n $, the null vector is $ \mathbf{0} = (0, 0, \dots, 0) $.

$$ \vec{0} = \begin{pmatrix} 0 \\ 0 \\ \vdots \\ 0 \end{pmatrix} $$

The Null Vector as the Additive Identity

Adding any vector $ \mathbf{v} $ to the null vector $ \mathbf{0} $ returns the original vector: $$ \mathbf{v} + \mathbf{0} = \mathbf{v} $$

This property follows directly from the definition of the null vector as the additive identity in a vector space.

Example

Consider a vector $ \vec{v} $ in the vector space $ \mathbb{R}^2 $:

$$ \vec{v} = \begin{pmatrix} 3 \\ -2 \end{pmatrix} $$

The null vector in this space is $ \vec{0} = \begin{pmatrix} 0 \\ 0 \end{pmatrix} $.

Adding the null vector to $ \vec{v} $ results in $ \vec{v} $:

$$ \vec{v} + \vec{0} = \begin{pmatrix} 3 \\ -2 \end{pmatrix} + \begin{pmatrix} 0 \\ 0 \end{pmatrix} $$

$$ \vec{v} + \vec{0} = \begin{pmatrix} 3 + 0 \\ -2 + 0 \end{pmatrix} $$

$$ \vec{v} + \vec{0} = \begin{pmatrix} 3 \\ -2 \end{pmatrix} $$

$$ \vec{v} + \vec{0} = \vec{v} $$

And so on.