What is \vec a + \vec b
?
\begin{align*}
\vec a &= AX \hat\imath + AY \hat\jmath \\
\vec b &= BX \hat\imath + BY \hat\jmath
\end{align*}
AX + BX \hat\imath + {}
AY + BY \hat\jmath
Sum the \hat\imath
and \hat\jmath
components separately.
\hat a + \hat b = (AX + BX) \hat\imath + (AY + BY) \hat\jmath
\hphantom{\hat a + \hat b} = AX + BX\hat\imath + AY + BY\hat\jmath