# Khan/khan-exercises

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randRangeNonZero( -9, 9 ) randRangeNonZero( -9, 9 ) randRangeNonZero( -9, 9 ) randRangeNonZero( -9, 9 ) randFromArray(["+", "-"]) SIGN === '+' ? [AX + BX, AY + BY] : [AX - BX, AY - BY]

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*}

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

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
line([AX, AY], SOLUTION, { stroke: GREEN, arrows: "->" }); var SF = 1 + 0.8 / sqrt(SOLUTION[0] * SOLUTION[0] + SOLUTION[1] * SOLUTION[1]); if (SIGN === '+') { label([SF * SOLUTION[0], SF * SOLUTION[1]], "\\vec b", { color: GREEN }); } else { label([SF * SOLUTION[0], SF * SOLUTION[1]], "-\\vec b", { color: GREEN }); }

\hphantom{\vec a + \vec b} = SOLUTION[0]\hat\imath + SOLUTION[1]\hat\jmath

line([0, 0], [SOLUTION[0], SOLUTION[1]], { stroke: RED, arrows: "->" });
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