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)

\(\displaystyle{v}={\left({v}_{{1}},{v}_{{2}},{v}_{{3}},{v}_{{4}}\right)}\)

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)