Define the two vectors as
[tex]\vec{a} = a_{x} \hat{i} + a_{y}\hat{j} + a_{z}\hat{k} \\ \vec{b}=b_{x}\hat{i}+b_{y}\hat{j}+b_{z}\hat{k}[/tex]
Then if vector c is defined by
[tex]\vec{c}=\vec{a}+\vec{b}[/tex]
then
[tex]\vec{c}=(a_{x}+b_{x})\hat{i} + (a_{y}+b_{y})\hat{j} + (a_{z}+b_{z})\hat{k}[/tex]
Example:
If
[tex]\vec{a}=3\hat{i}-7\hat{j} \\ \vec{b}=-2\hat{i}+5\hat{k}[/tex]
then
[tex]\vec{c}=(3-2)\hat{i}+(-7+0)\hat{j}+(0+5)\hat{k} \\ \vec{c}=\hat{i}-7\hat{j}+5\hat{k}[/tex]
