# Describe the zero vector (the additive identity) of the vector

Describe the zero vector (the additive identity) of the vector space.
$$\displaystyle{R}^{{4}}$$

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Knes1997
Every vector in $$\displaystyle{\mathbb{{{R}^{{4}}}}}$$ can be represented as
$$\displaystyle{v}={\left({v}_{{1}},{v}_{{2}},{v}_{{3}},{v}_{{4}}\right)}$$
We have to find the additive identity vector - the vector that has following property:
$$\displaystyle{v}+{x}={x}+{v}={v}$$
In coordinate notation:
$$\displaystyle{\left({v}_{{1}},{v}_{{2}},{v}_{{3}},{v}_{{3}},{v}_{{4}}\right)}$$
We have to find the additive identity vector - the vector that has following property:
$$\displaystyle{v}+{x}={x}+{v}={v}$$
In coordinate notation:
$$\displaystyle{\left({v}_{{1}},{v}_{{2}},{v}_{{3}},{v}_{{4}}\right)}+{\left({x}_{{1}},{x}_{{2}},{x}_{{3}},{x}_{{4}}\right)}={\left({v}_{{1}}+{x}_{{1}},{v}_{{2}}+{x}_{{2}},{v}_{{3}}+{x}_{{3}},{v}_{{4}}+{x}_{{4}}\right)}={\left({v}_{{1}},{v}_{{2}},{v}_{{3}},{v}_{{4}}\right)}$$
$$\displaystyle{v}_{{1}}+{x}_{{1}}={v}_{{1}}$$
$$\displaystyle{v}_{{2}}+{x}_{{2}}={v}_{{2}}$$
$$\displaystyle{v}_{{3}}+{x}_{{3}}={v}_{{3}}$$
$$\displaystyle{v}_{{4}}+{x}_{{4}}={v}_{{4}}$$
From properties of addition in $$\displaystyle{\mathbb{{{R}}}}$$, we know that $$\displaystyle{x}_{{1}}={x}_{{2}}={x}_{{3}}={x}_{{4}}={0}$$
So, additive identity vector is $$\displaystyle{x}={\left({0},{0},{0},{0}\right)}$$.
Result: (0,0,0,0)