# Describe the zero vector (the additive identity) of the vector space. R^4

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