randRangeNonZero(-3, 3) randRangeNonZero(-3, 3) randFromArray([-1, -0.5, 0.5, -2, 2, -3, 3]) randRangeNonZero(-7, 7) randRangeNonZero(-7, 7) randRangeNonZero(-7, 7) randRangeNonZero(-7, 7) AX * SA AY * SA -AX * SA -AY * SA [DX, DY] shuffle([[BX, BY], [CX, CY], ANS, [EX, EY]]) [ ["b", "pink"], ["c", "green"], ["d", "purple"], ["e", "red"] ][ $.inArray( ANS, SHUF ) ] SHUF[0] SHUF[1] SHUF[2] SHUF[3] randRangeNonZero(max(-9, -9 - AX), min(9, 9 - AX)) randRangeNonZero(max(-9, -9 - AY), min(9, 9 - AY)) randRangeNonZero(max(-9, -9 - BX), min(9, 9 - BX)) randRangeNonZero(max(-9, -9 - BY), min(9, 9 - BY)) randRangeNonZero(max(-9, -9 - CX), min(9, 9 - CX)) randRangeNonZero(max(-9, -9 - CY), min(9, 9 - CY)) randRangeNonZero(max(-9, -9 - DX), min(9, 9 - DX)) randRangeNonZero(max(-9, -9 - DY), min(9, 9 - DY)) randRangeNonZero(max(-9, -9 - EX), min(9, 9 - EX)) randRangeNonZero(max(-9, -9 - EY), min(9, 9 - EY)) 1 + 0.8 / sqrt(AX * AX + AY * AY) 1 + 0.8 / sqrt(BX * BX + BY * BY) 1 + 0.8 / sqrt(CX * CX + CY * CY) 1 + 0.8 / sqrt(DX * DX + DY * DY) 1 + 0.8 / sqrt(EX * EX + EY * EY)

What is `-\vec a`?

What is `decimalFraction(SA, true) \vec a`?

graphInit({ range: 10, scale: 20, tickStep: 1, axisArrows: "<->" }); style({ stroke: BLUE, color: BLUE }, function() { line( [AOX, AOY], [AOX + AX, AOY + AY], { arrows: "->" } ); label( [AOX + AF * AX, AOY + AF * AY], "\\vec a" ); }); style({ stroke: PINK, color: PINK }, function() { line( [BOX, BOY], [BOX + BX, BOY + BY], { arrows: "->" } ); label( [BOX + BF * BX, BOY + BF * BY], "\\vec b" ); }); style({ stroke: GREEN, color: GREEN }, function() { line( [COX, COY], [COX + CX, COY + CY], { arrows: "->" } ); label( [COX + CF * CX, COY + CF * CY], "\\vec c" ); }); style({ stroke: PURPLE, color: PURPLE }, function() { line( [DOX, DOY], [DOX + DX, DOY + DY], { arrows: "->" } ); label( [DOX + DF * DX, DOY + DF * DY], "\\vec d" ); }); style({ stroke: RED, color: RED }, function() { line( [EOX, EOY], [EOX + EX, EOY + EY], { arrows: "->" } ); label( [EOX + EF * EX, EOY + EF * EY], "\\vec e" ); });

`\large\ANSC{\vec ANSL}`

- `\large\pink{\vec b}`
- `\large\green{\vec c}`
- `\large\purple{\vec d}`
- `\large\red{\vec e}`

Reading from the graph, we see that `\vec a = AX \hat\imath + AY \hat\jmath`.

`SA \vec a = SA \cdot (AX \hat\imath + AY \hat\jmath)`.

`\hphantom{SA \vec a} = (SA \cdot AX) \hat\imath + (SA \cdot AY) \hat\jmath`.

`\hphantom{SA \vec a} = SA * AX \hat\imath + SA * AY \hat\jmath`.

The vector that matches is `\vec ANSL`.

randRangeNonZero(-8, 8) randRangeNonZero(-8, 8) randFromArray([-1, -0.5, 0.5, -2, 2, -3, 3, -4, 4, -5, 5])

`\vec a = AX \hat\imath + AY \hat\jmath`.

What is `-\vec a`?

What is `decimalFraction(SA, true) \vec a`?

AX * SA `\hat\imath + {}` AY * SA `\hat\jmath`

`SA \vec a = SA \cdot (AX \hat\imath + AY \hat\jmath)`

`\hphantom{SA \vec a} = (SA \cdot AX) \hat\imath + (SA \cdot AY) \hat\jmath`

`\hphantom{SA \vec a} = SA * AX \hat\imath + SA * AY \hat\jmath`

randRangeNonZero(-8, 8) randRangeNonZero(-8, 8) randFromArray([-1, -0.5, 0.5, -2, 2, -3, 3, -4, 4, -5, 5])

`\vec a = (AX, AY)`

What is `-\vec a`?

What is `decimalFraction(SA, true) \vec a`?

`-\vec adecimalFraction(SA, true) \vec a = (` AX * SA, AY * SA`)`

`SA \vec a = SA \cdot (AX, AY)`

`\hphantom{SA \vec a} = (SA \cdot AX, SA \cdot AY)`

`\hphantom{SA \vec a} = (SA * AX, SA * AY)`

randFromArray([-1, -0.5, 0.5, -2, 2, -0.25, 0.25, -4, 4])
abs(SA) < 1 ? randRangeNonZero(-4, 4) * 2 : randRangeNonZero(-4, 4) / 2 abs(SA) < 1 ? randRangeNonZero(-4, 4) * 2 : randRangeNonZero(-4, 4) / 2
1 + 0.8 / sqrt(AX * AX + AY * AY) 1 + 0.8 / (abs(SA) * sqrt(AX * AX + AY * AY)) [[AX], [AY]] randFromArray([ AX + "\\hat\\imath + " + AY + "\\hat\\jmath", "(" + AX + ", " + AY + ")" ]) $._("against") $._("along") $._("positive") $._("negative") ["", AGAINST, ALONG] SA < 0 ? AGAINST : ALONG SA < 0 ? NEGATIVE : POSITIVE

`\vec a = FORMAT`

`\vec b = -\vec a`

`\vec b = decimalFraction(SA, true) \vec a`

What is the magnitude and direction of `\vec b` compared to `\vec a`?

`\vec b` is abs(SA) times the size of `\vec a`.
`\vec b` moves SOLUTION `\vec a`.

graphInit({ range: 10, scale: 20, tickStep: 1, axisArrows: "<->" }); style({ stroke: BLUE, color: BLUE }, function() { line([0, 0], [AX, AY], { arrows: "->" } ); label([AF * AX, AF * AY], "\\vec a" ); });
style({ strokeDasharray: ".", color: BLACK }, function() { line([0, 0], [AX * SA, AY * SA], { arrows: "->" } ); label([BF * AX * SA, BF * AY * SA], "\\vec b" ); });

`\vec a` is scaled by `SA` to create vector `\vec b`.

`SA` is SIGN, so the direction of `\vec b` is SOLUTION `\vec a`.

Each of the components of `\vec a` is multiplied by `SA` to get `\vec b`, so the magnitude of `\vec b` is `abs(SA)` times the size of `\vec a`.