randRangeNonZero( -9, 9 ) randRangeNonZero( -9, 9 ) randRangeNonZero( -9, 9 ) randRangeNonZero( -9, 9 )

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*}

graphInit({ range: 10, scale: 20, tickStep: 1, labelStep: 1, unityLabels: false, labelFormat: function( s ) { return "\\small{" + s + "}"; }, axisArrows: "<->" }); line( [0, 0], [AX, AY], { stroke: "#6495ed", arrows: "->" } ); line( [0, 0], [BX, BY], { stroke: "#28ae7b", arrows: "->" } ); var AF = 1 + 0.8 / sqrt( AX * AX + AY * AY ); label( [AF * AX, AF * AY], "\\vec a", { color: "#6495ed" } ); var BF = 1 + 0.8 / sqrt( BX * BX + BY * BY ); label( [BF * BX, BF * BY], "\\vec b", { color: "#28ae7b" } );

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

line( [BX, BY], [BX + AX, BY + AY], { stroke: "#6495ed", arrows: "->" } );

\hphantom{\hat a + \hat b} = AX + BX\hat\imath + AY + BY\hat\jmath

line( [0, 0], [BX + AX, BY + AY], { stroke: "#ffa500", arrows: "->" } );