What is \vec a SIGN \vec b?
\begin{align*}
\vec a &= AX \hat\imath + AY \hat\jmath \\
\vec b &= BX \hat\imath + BY \hat\jmath
\end{align*}
SOLUTION[0]\hat\imath + {}
SOLUTION[1] \hat\jmath
Sum the \hat\imath and \hat\jmath components separately.
\blue{\vec a} SIGN \green{\vec b} =
\blue{(AX \hat\imath + AY \hat\jmath)} SIGN
\green{(BX \hat\imath + BY \hat\jmath)}
\hphantom{\vec a + \vec b} =
\blue{AX \hat\imath + AY \hat\jmath}
+-
\green{abs(BX) \hat\imath + BY \hat\jmath}
\hphantom{\vec a + \vec b} =
\blue{AX \hat\imath + AY \hat\jmath}
-+
\green{abs(BX) \hat\imath + -BY \hat\jmath}
\hphantom{\vec a + \vec b} = (\blue{AX}
+-
\green{abs(BX)}) \hat\imath + (\blue{AY}
+-
\green{abs(BY)}) \hat\jmath
\hphantom{\vec a + \vec b} = (\blue{AX}
-+
\green{abs(BX)}) \hat\imath + (\blue{AY}
-+
\green{abs(BY)}) \hat\jmath
\hphantom{\vec a + \vec b} = SOLUTION[0]\hat\imath + SOLUTION[1]\hat\jmath