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False, $CellContext`d1$$ = 4, $CellContext`d2$$ = 1, $CellContext`d3$$ = 3, $CellContext`vpoint$$ = {-0.5, 0.7}, $CellContext`zoom$$ = 1., Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{ Hold[ Column[{ Row[{"examples:", ButtonBar[{ " random " :> ({{$CellContext`a1$$, $CellContext`b1$$, \ $CellContext`c1$$, $CellContext`d1$$}, {$CellContext`a2$$, $CellContext`b2$$, \ $CellContext`c2$$, $CellContext`d2$$}, {$CellContext`a3$$, $CellContext`b3$$, \ $CellContext`c3$$, $CellContext`d3$$}} = $CellContext`getExamples[1]), " orthogonal " :> ({{$CellContext`a1$$, $CellContext`b1$$, \ $CellContext`c1$$, $CellContext`d1$$}, {$CellContext`a2$$, $CellContext`b2$$, \ $CellContext`c2$$, $CellContext`d2$$}, {$CellContext`a3$$, $CellContext`b3$$, \ $CellContext`c3$$, $CellContext`d3$$}} = $CellContext`getExamples[2]), " line " :> ({{$CellContext`a1$$, $CellContext`b1$$, \ $CellContext`c1$$, $CellContext`d1$$}, {$CellContext`a2$$, $CellContext`b2$$, \ $CellContext`c2$$, $CellContext`d2$$}, {$CellContext`a3$$, $CellContext`b3$$, \ $CellContext`c3$$, $CellContext`d3$$}} = $CellContext`getExamples[3]), " plane " :> ({{$CellContext`a1$$, $CellContext`b1$$, \ $CellContext`c1$$, $CellContext`d1$$}, {$CellContext`a2$$, $CellContext`b2$$, \ $CellContext`c2$$, $CellContext`d2$$}, {$CellContext`a3$$, $CellContext`b3$$, \ $CellContext`c3$$, $CellContext`d3$$}} = $CellContext`getExamples[4]), " no solution " :> ({{$CellContext`a1$$, $CellContext`b1$$, \ $CellContext`c1$$, $CellContext`d1$$}, {$CellContext`a2$$, $CellContext`b2$$, \ $CellContext`c2$$, $CellContext`d2$$}, {$CellContext`a3$$, $CellContext`b3$$, \ $CellContext`c3$$, $CellContext`d3$$}} = $CellContext`getExamples[5]), " default " :> ({{$CellContext`a1$$, $CellContext`b1$$, \ $CellContext`c1$$, $CellContext`d1$$}, {$CellContext`a2$$, $CellContext`b2$$, \ $CellContext`c2$$, $CellContext`d2$$}, {$CellContext`a3$$, $CellContext`b3$$, \ $CellContext`c3$$, $CellContext`d3$$}} = $CellContext`getExamples[6])}]}], Framed[ OpenerView[{"or select your own equation parameters", Column[{ Row[{ Manipulate`Place[1], Manipulate`Place[2], Manipulate`Place[3], Manipulate`Place[4], Manipulate`Place[5]}], Row[{ Manipulate`Place[6], Manipulate`Place[7], Manipulate`Place[8], Manipulate`Place[9], Manipulate`Place[10]}], Row[{ Manipulate`Place[11], Manipulate`Place[12], Manipulate`Place[13], Manipulate`Place[14], Manipulate`Place[15]}]}, Center]}], RoundingRadius -> 5, FrameMargins -> 8], Row[{ Manipulate`Place[16], Manipulate`Place[17], Manipulate`Place[18], Button[ Style[ "reset view", 10], {$CellContext`vpoint$$, $CellContext`zoom$$} = {{-0.5, 0.7}, 1.}, Appearance -> "Palette"]}, Spacer[4]]}, Center, ItemSize -> 55]], Manipulate`Dump`ThisIsNotAControl}, {{ Hold[$CellContext`col1$$], RGBColor[0.33, 0.67, 0.7], " "}, GrayLevel[0.5]}, {{ Hold[$CellContext`a1$$], 1, " "}, {-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}}, {{ Hold[$CellContext`b1$$], 2, Style[" x +", 14]}, {-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}}, {{ Hold[$CellContext`c1$$], 3, Style[" y +", 14]}, {-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}}, {{ Hold[$CellContext`d1$$], 4, Style[" z \[Equal]", 14]}, {-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}}, {{ Hold[$CellContext`col2$$], RGBColor[0.79, 0.71, 0.26], " "}, GrayLevel[0.5]}, {{ Hold[$CellContext`a2$$], 4, " "}, {-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}}, {{ Hold[$CellContext`b2$$], 3, Style[" x +", 14]}, {-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}}, {{ Hold[$CellContext`c2$$], 2, Style[" y +", 14]}, {-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}}, {{ Hold[$CellContext`d2$$], 1, Style[" z \[Equal]", 14]}, {-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}}, {{ Hold[$CellContext`col3$$], RGBColor[0.48, 0.11, 0.22], " "}, GrayLevel[0.5]}, {{ Hold[$CellContext`a3$$], -1, " "}, {-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}}, {{ Hold[$CellContext`b3$$], -2, Style[" x +", 14]}, {-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}}, {{ Hold[$CellContext`c3$$], -1, Style[" y +", 14]}, {-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}}, {{ Hold[$CellContext`d3$$], 3, Style[" z \[Equal]", 14]}, {-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}}, {{ Hold[$CellContext`columnInterpretationQ$$], False, ""}, { False -> "row view", True -> "column view"}}, {{ Hold[$CellContext`vpoint$$], {-0.5, 0.7}, Style["view", Bold]}, {-Pi, Rational[-1, 2] Pi}, { Pi, Rational[1, 2] Pi}}, {{ Hold[$CellContext`zoom$$], 1., Style["zoom", Bold]}, 0.2, 2.5}}, Typeset`size$$ = { 500., {172.5, 177.5}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = False, $CellContext`a1$14250$$ = 0, $CellContext`b1$14251$$ = 0, $CellContext`c1$14252$$ = 0, $CellContext`d1$14253$$ = 0, $CellContext`a2$14254$$ = 0, $CellContext`b2$14255$$ = 0, $CellContext`c2$14256$$ = 0, $CellContext`d2$14257$$ = 0, $CellContext`columnInterpretationQ$14258$$ = False, $CellContext`vpoint$14259$$ = {0, 0}}, DynamicBox[Manipulate`ManipulateBoxes[ 2, StandardForm, "Variables" :> {$CellContext`a1$$ = 1, $CellContext`a2$$ = 4, $CellContext`a3$$ = -1, $CellContext`b1$$ = 2, $CellContext`b2$$ = 3, $CellContext`b3$$ = -2, $CellContext`c1$$ = 3, $CellContext`c2$$ = 2, $CellContext`c3$$ = -1, $CellContext`col1$$ = RGBColor[0.33, 0.67, 0.7], $CellContext`col2$$ = RGBColor[0.79, 0.71, 0.26], $CellContext`col3$$ = RGBColor[0.48, 0.11, 0.22], $CellContext`columnInterpretationQ$$ = False, $CellContext`d1$$ = 4, $CellContext`d2$$ = 1, $CellContext`d3$$ = 3, $CellContext`vpoint$$ = {-0.5, 0.7}, $CellContext`zoom$$ = 1.}, "ControllerVariables" :> { Hold[$CellContext`a1$$, $CellContext`a1$14250$$, 0], Hold[$CellContext`b1$$, $CellContext`b1$14251$$, 0], Hold[$CellContext`c1$$, $CellContext`c1$14252$$, 0], Hold[$CellContext`d1$$, $CellContext`d1$14253$$, 0], Hold[$CellContext`a2$$, $CellContext`a2$14254$$, 0], Hold[$CellContext`b2$$, $CellContext`b2$14255$$, 0], Hold[$CellContext`c2$$, $CellContext`c2$14256$$, 0], Hold[$CellContext`d2$$, $CellContext`d2$14257$$, 0], Hold[$CellContext`columnInterpretationQ$$, \ $CellContext`columnInterpretationQ$14258$$, False], Hold[$CellContext`vpoint$$, $CellContext`vpoint$14259$$, {0, 0}]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> Module[{$CellContext`colRange$, $CellContext`rowRange$, \ $CellContext`aRank$, $CellContext`xSol$, $CellContext`ySol$, \ $CellContext`zSol$, $CellContext`solFlag$, $CellContext`varX$, \ $CellContext`varY$, $CellContext`varZ$}, {$CellContext`xSol$, \ $CellContext`ySol$, $CellContext`zSol$, $CellContext`solFlag$} = \ $CellContext`SolveSystem[{{$CellContext`a1$$, $CellContext`b1$$, \ $CellContext`c1$$, $CellContext`d1$$}, {$CellContext`a2$$, $CellContext`b2$$, \ $CellContext`c2$$, $CellContext`d2$$}, {$CellContext`a3$$, $CellContext`b3$$, \ $CellContext`c3$$, $CellContext`d3$$}}]; $CellContext`aRank$ = MatrixRank[{{$CellContext`a1$$, $CellContext`b1$$, \ $CellContext`c1$$}, {$CellContext`a2$$, $CellContext`b2$$, \ $CellContext`c2$$}, {$CellContext`a3$$, $CellContext`b3$$, \ $CellContext`c3$$}}]; $CellContext`colRange$ = Max[{2, Max[ Abs[{$CellContext`a1$$, $CellContext`a2$$, $CellContext`a3$$, \ $CellContext`b1$$, $CellContext`b2$$, $CellContext`b3$$, $CellContext`c1$$, \ $CellContext`c2$$, $CellContext`c3$$, $CellContext`d1$$, $CellContext`d2$$, \ $CellContext`d3$$}]]}]; $CellContext`rowRange$ = If[$CellContext`solFlag$ == 3, Min[{100, Max[20, Max[ Abs[{$CellContext`xSol$, $CellContext`ySol$, \ $CellContext`zSol$}]]]}], 20]; Pane[ Text[ Column[{ If[$CellContext`columnInterpretationQ$$, Row[{ $CellContext`ColInterpFormatter[{{$CellContext`a1$$, \ $CellContext`b1$$, $CellContext`c1$$, $CellContext`d1$$}, {$CellContext`a2$$, \ $CellContext`b2$$, $CellContext`c2$$, $CellContext`d2$$}, {$CellContext`a3$$, \ $CellContext`b3$$, $CellContext`c3$$, $CellContext`d3$$}}, 13], " solution: ", $CellContext`solutionFormatter[{$CellContext`xSol$, \ $CellContext`ySol$, $CellContext`zSol$, $CellContext`solFlag$}, 13]}], Row[{ $CellContext`sysFormatterColor[{{$CellContext`a1$$, \ $CellContext`b1$$, $CellContext`c1$$, $CellContext`d1$$}, {$CellContext`a2$$, \ $CellContext`b2$$, $CellContext`c2$$, $CellContext`d2$$}, {$CellContext`a3$$, \ $CellContext`b3$$, $CellContext`c3$$, $CellContext`d3$$}}, \ {$CellContext`col1$$, $CellContext`col2$$, $CellContext`col3$$}, 13], " solution: ", $CellContext`solutionFormatter[{$CellContext`xSol$, \ $CellContext`ySol$, $CellContext`zSol$, $CellContext`solFlag$}, 13]}]], Row[{ Switch[$CellContext`aRank$, 0, Row[{"Matrix ", Style["A", Italic], " has all zeros."}], 1, Row[{"Columns of ", Style["A", Italic], " are collinear."}], 2, Row[{"Columns of ", Style["A", Italic], " are coplanar."}], 3, Row[{"Columns of ", Style["A", Italic], " are independent."}]], If[ And[$CellContext`aRank$ == 1, $CellContext`solFlag$ == 1], " Vector b lies on the same line."], If[ And[$CellContext`aRank$ == 2, $CellContext`solFlag$ == 2], Row[{"Vector ", Style["b", Italic], " lies on the same plane."}]]}], If[$CellContext`columnInterpretationQ$$, $CellContext`columnVectors[{$CellContext`a1$$, \ $CellContext`a2$$, $CellContext`a3$$}, {$CellContext`b1$$, $CellContext`b2$$, \ $CellContext`b3$$}, {$CellContext`c1$$, $CellContext`c2$$, \ $CellContext`c3$$}, {$CellContext`d1$$, $CellContext`d2$$, \ $CellContext`d3$$}, $CellContext`col1$$, $CellContext`col2$$, \ $CellContext`col3$$, $CellContext`colRange$, -Part[$CellContext`vpoint$$, 1], Part[$CellContext`vpoint$$, 2], $CellContext`solFlag$, $CellContext`xSol$, \ $CellContext`ySol$, $CellContext`zSol$, 1/$CellContext`zoom$$], $CellContext`rowView[{$CellContext`xSol$, $CellContext`ySol$, \ $CellContext`zSol$}, $CellContext`solFlag$, {{$CellContext`a1$$, \ $CellContext`b1$$, $CellContext`c1$$, $CellContext`d1$$}, {$CellContext`a2$$, \ $CellContext`b2$$, $CellContext`c2$$, $CellContext`d2$$}, {$CellContext`a3$$, \ $CellContext`b3$$, $CellContext`c3$$, $CellContext`d3$$}}, \ $CellContext`col1$$, $CellContext`col2$$, $CellContext`col3$$, \ $CellContext`rowRange$, -Part[$CellContext`vpoint$$, 1], Part[$CellContext`vpoint$$, 2], 1/$CellContext`zoom$$]]}, Center]], {Automatic, 350}]], "Specifications" :> { Column[{ Row[{"examples:", ButtonBar[{ " random " :> ({{$CellContext`a1$$, $CellContext`b1$$, \ $CellContext`c1$$, $CellContext`d1$$}, {$CellContext`a2$$, $CellContext`b2$$, \ $CellContext`c2$$, $CellContext`d2$$}, {$CellContext`a3$$, $CellContext`b3$$, \ $CellContext`c3$$, $CellContext`d3$$}} = $CellContext`getExamples[1]), " orthogonal " :> ({{$CellContext`a1$$, $CellContext`b1$$, \ $CellContext`c1$$, $CellContext`d1$$}, {$CellContext`a2$$, $CellContext`b2$$, \ $CellContext`c2$$, $CellContext`d2$$}, {$CellContext`a3$$, $CellContext`b3$$, \ $CellContext`c3$$, $CellContext`d3$$}} = $CellContext`getExamples[2]), " line " :> ({{$CellContext`a1$$, $CellContext`b1$$, \ $CellContext`c1$$, $CellContext`d1$$}, {$CellContext`a2$$, $CellContext`b2$$, \ $CellContext`c2$$, $CellContext`d2$$}, {$CellContext`a3$$, $CellContext`b3$$, \ $CellContext`c3$$, $CellContext`d3$$}} = $CellContext`getExamples[3]), " plane " :> ({{$CellContext`a1$$, $CellContext`b1$$, \ $CellContext`c1$$, $CellContext`d1$$}, {$CellContext`a2$$, $CellContext`b2$$, \ $CellContext`c2$$, $CellContext`d2$$}, {$CellContext`a3$$, $CellContext`b3$$, \ $CellContext`c3$$, $CellContext`d3$$}} = $CellContext`getExamples[4]), " no solution " :> ({{$CellContext`a1$$, $CellContext`b1$$, \ $CellContext`c1$$, $CellContext`d1$$}, {$CellContext`a2$$, $CellContext`b2$$, \ $CellContext`c2$$, $CellContext`d2$$}, {$CellContext`a3$$, $CellContext`b3$$, \ $CellContext`c3$$, $CellContext`d3$$}} = $CellContext`getExamples[5]), " default " :> ({{$CellContext`a1$$, $CellContext`b1$$, \ $CellContext`c1$$, $CellContext`d1$$}, {$CellContext`a2$$, $CellContext`b2$$, \ $CellContext`c2$$, $CellContext`d2$$}, {$CellContext`a3$$, $CellContext`b3$$, \ $CellContext`c3$$, $CellContext`d3$$}} = $CellContext`getExamples[6])}]}], Framed[ OpenerView[{"or select your own equation parameters", Column[{ Row[{ Manipulate`Place[1], Manipulate`Place[2], Manipulate`Place[3], Manipulate`Place[4], Manipulate`Place[5]}], Row[{ Manipulate`Place[6], Manipulate`Place[7], Manipulate`Place[8], Manipulate`Place[9], Manipulate`Place[10]}], Row[{ Manipulate`Place[11], Manipulate`Place[12], Manipulate`Place[13], Manipulate`Place[14], Manipulate`Place[15]}]}, Center]}], RoundingRadius -> 5, FrameMargins -> 8], Row[{ Manipulate`Place[16], Manipulate`Place[17], Manipulate`Place[18], Button[ Style[ "reset view", 10], {$CellContext`vpoint$$, $CellContext`zoom$$} = {{-0.5, 0.7}, 1.}, Appearance -> "Palette"]}, Spacer[4]]}, Center, ItemSize -> 55], {{$CellContext`col1$$, RGBColor[0.33, 0.67, 0.7], " "}, GrayLevel[0.5], ControlType -> ColorSetter, ControlPlacement -> 1}, {{$CellContext`a1$$, 1, " "}, {-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, Appearance -> "Palette", Alignment -> Right, ControlPlacement -> 2}, {{$CellContext`b1$$, 2, Style[" x +", 14]}, {-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, Appearance -> "Palette", Alignment -> Right, ControlPlacement -> 3}, {{$CellContext`c1$$, 3, Style[" y +", 14]}, {-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, Appearance -> "Palette", Alignment -> Right, ControlPlacement -> 4}, {{$CellContext`d1$$, 4, Style[" z \[Equal]", 14]}, {-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, Appearance -> "Palette", Alignment -> Right, ControlPlacement -> 5}, {{$CellContext`col2$$, RGBColor[0.79, 0.71, 0.26], " "}, GrayLevel[0.5], ControlType -> ColorSetter, ControlPlacement -> 6}, {{$CellContext`a2$$, 4, " "}, {-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, Appearance -> "Palette", Alignment -> Right, ControlPlacement -> 7}, {{$CellContext`b2$$, 3, Style[" x +", 14]}, {-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, Appearance -> "Palette", Alignment -> Right, ControlPlacement -> 8}, {{$CellContext`c2$$, 2, Style[" y +", 14]}, {-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, Appearance -> "Palette", Alignment -> Right, ControlPlacement -> 9}, {{$CellContext`d2$$, 1, Style[" z \[Equal]", 14]}, {-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, Appearance -> "Palette", Alignment -> Right, ControlPlacement -> 10}, {{$CellContext`col3$$, RGBColor[0.48, 0.11, 0.22], " "}, GrayLevel[0.5], ControlType -> ColorSetter, ControlPlacement -> 11}, {{$CellContext`a3$$, -1, " "}, {-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, Appearance -> "Palette", Alignment -> Right, ControlPlacement -> 12}, {{$CellContext`b3$$, -2, Style[" x +", 14]}, {-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, Appearance -> "Palette", Alignment -> Right, ControlPlacement -> 13}, {{$CellContext`c3$$, -1, Style[" y +", 14]}, {-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, Appearance -> "Palette", Alignment -> Right, ControlPlacement -> 14}, {{$CellContext`d3$$, 3, Style[" z \[Equal]", 14]}, {-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, Appearance -> "Palette", Alignment -> Right, ControlPlacement -> 15}, {{$CellContext`columnInterpretationQ$$, False, ""}, { False -> "row view", True -> "column view"}, Appearance -> Small, ControlPlacement -> 16}, {{$CellContext`vpoint$$, {-0.5, 0.7}, Style["view", Bold]}, {-Pi, Rational[-1, 2] Pi}, { Pi, Rational[1, 2] Pi}, Appearance -> Tiny, ImageSize -> Small, ControlPlacement -> 17}, {{$CellContext`zoom$$, 1., Style["zoom", Bold]}, 0.2, 2.5, Appearance -> Tiny, ImageSize -> Tiny, ControlPlacement -> 18}}, "Options" :> {Deployed -> True, AutorunSequencing -> {16, 17}}, "DefaultOptions" :> {ControllerLinking -> True}], ImageSizeCache->{545., {265., 270.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, Initialization:>(({$CellContext`SolveSystem[ Pattern[$CellContext`myMat, Blank[List]]] := Module[{$CellContext`solX, $CellContext`solY, $CellContext`solZ, \ $CellContext`tempSol, $CellContext`solFlag}, {$CellContext`tempSol, \ $CellContext`solFlag} = $CellContext`SolveSystem2[$CellContext`myMat]; \ {$CellContext`solX, $CellContext`solY, $CellContext`solZ} = ReplaceAll[{$CellContext`varX, $CellContext`varY, \ $CellContext`varZ}, $CellContext`tempSol]; {$CellContext`solX, \ $CellContext`solY, $CellContext`solZ, $CellContext`solFlag}], \ $CellContext`SolveSystem2[ Pattern[$CellContext`myMat, Blank[List]]] := Quiet[ Module[{$CellContext`tempSol, $CellContext`solFlag, $CellContext`x, \ $CellContext`y, $CellContext`z}, If[Norm[$CellContext`myMat] == 0, $CellContext`tempSol = {}; $CellContext`solFlag = 4; Null, $CellContext`tempSol = Part[ Solve[{Part[$CellContext`myMat, 1, 1] $CellContext`varX + Part[$CellContext`myMat, 1, 2] $CellContext`varY + Part[$CellContext`myMat, 1, 3] $CellContext`varZ == Part[$CellContext`myMat, 1, 4], Part[$CellContext`myMat, 2, 1] $CellContext`varX + Part[$CellContext`myMat, 2, 2] $CellContext`varY + Part[$CellContext`myMat, 2, 3] $CellContext`varZ == Part[$CellContext`myMat, 2, 4], Part[$CellContext`myMat, 3, 1] $CellContext`varX + Part[$CellContext`myMat, 3, 2] $CellContext`varY + Part[$CellContext`myMat, 3, 3] $CellContext`varZ == Part[$CellContext`myMat, 3, 4]}, {$CellContext`varX, $CellContext`varY, \ $CellContext`varZ}], 1]; $CellContext`solFlag = 0; $CellContext`solFlag = Length[$CellContext`tempSol]; If[ Not[ StringFreeQ[ ToString[$CellContext`tempSol], "{}"]], $CellContext`solFlag = 0; $CellContext`tempSol = {}]; Null]; {$CellContext`tempSol, $CellContext`solFlag}]], \ $CellContext`ColInterpFormatter[{{ Pattern[$CellContext`a1, Blank[]], Pattern[$CellContext`b1, Blank[]], Pattern[$CellContext`c1, Blank[]], Pattern[$CellContext`d1, Blank[]]}, { Pattern[$CellContext`a2, Blank[]], Pattern[$CellContext`b2, Blank[]], Pattern[$CellContext`c2, Blank[]], Pattern[$CellContext`d2, Blank[]]}, { Pattern[$CellContext`a3, Blank[]], Pattern[$CellContext`b3, Blank[]], Pattern[$CellContext`c3, Blank[]], Pattern[$CellContext`d3, Blank[]]}}, Optional[ Pattern[$CellContext`size, Blank[]], 12]] := Row[ Map[TraditionalForm, { Row[{ Style["x", Italic], " \[CenterDot]"}], MatrixForm[{{ Text[ Style[$CellContext`a1, $CellContext`size, Green, Bold]]}, { Text[ Style[$CellContext`a2, $CellContext`size, Green, Bold]]}, { Text[ Style[$CellContext`a3, $CellContext`size, Green, Bold]]}}], Row[{" + ", Style["y", Italic], " \[CenterDot]"}], MatrixForm[{{ Text[ Style[$CellContext`b1, $CellContext`size, Orange, Bold]]}, { Text[ Style[$CellContext`b2, $CellContext`size, Orange, Bold]]}, { Text[ Style[$CellContext`b3, $CellContext`size, Orange, Bold]]}}], Row[{" + ", Style["z", Italic], " \[CenterDot]"}], MatrixForm[{{ Text[ Style[$CellContext`c1, $CellContext`size, Blue, Bold]]}, { Text[ Style[$CellContext`c2, $CellContext`size, Blue, Bold]]}, { Text[ Style[$CellContext`c3, $CellContext`size, Blue, Bold]]}}], " = ", MatrixForm[{{ Text[ Style[$CellContext`d1, $CellContext`size, Red, Bold]]}, { Text[ Style[$CellContext`d2, $CellContext`size, Red, Bold]]}, { Text[ Style[$CellContext`d3, $CellContext`size, Red, Bold]]}}]}]], $CellContext`solutionFormatter[{ Pattern[$CellContext`xsol, Blank[]], Pattern[$CellContext`ysol, Blank[]], Pattern[$CellContext`zsol, Blank[]], Pattern[$CellContext`solflag, Blank[]]}, Optional[ Pattern[$CellContext`size, Blank[]], 12]] := Switch[$CellContext`solflag, 0, Text[ Style["NO SOLUTION", $CellContext`size, Red]], 1, $CellContext`solFormatter[{$CellContext`xsol, $CellContext`ysol, \ $CellContext`zsol}, $CellContext`size], 2, $CellContext`solFormatter[{$CellContext`xsol, $CellContext`ysol, \ $CellContext`zsol}, $CellContext`size], 3, $CellContext`solFormatter[{$CellContext`xsol, $CellContext`ysol, \ $CellContext`zsol}, $CellContext`size], 4, Text[ Style[ Row[{"\[ForAll] {", Style["x", Italic], ",", Style["y", Italic], ",", Style["z", Italic], "} \[Element] \!\(\*SuperscriptBox[\(\[DoubleStruckCapitalR]\), \ \(3\)]\)"}], $CellContext`size, Red]]], $CellContext`solFormatter[{ Pattern[$CellContext`xsol, Blank[]], Pattern[$CellContext`ysol, Blank[]], Pattern[$CellContext`zsol, Blank[]]}, Optional[ Pattern[$CellContext`size, Blank[]], 12]] := TableForm[ Map[Style[#, $CellContext`size, Black]& , { If[ToString[$CellContext`xsol] == "varX", " ", Row[{ Style["x", Italic], " = ", StringReplace[ ToString[ TraditionalForm[$CellContext`xsol]], { "varY" -> "y", "varZ" -> "z"}]}]], " ", If[ToString[$CellContext`ysol] == "varY", " ", Row[{ Style["y", Italic], " = ", StringReplace[ ToString[ TraditionalForm[$CellContext`ysol]], { "varX" -> "x", "varZ" -> "z"}]}]], " ", If[ToString[$CellContext`zsol] == "varZ", " ", Row[{ Style["z", Italic], " = ", StringReplace[ ToString[ TraditionalForm[$CellContext`zsol]], { "varY" -> "y", "varX" -> "x"}]}]]}], TableSpacing -> {0, 0}, TableDirections -> Column], $CellContext`sysFormatterColor[{{ Pattern[$CellContext`a1, Blank[]], Pattern[$CellContext`b1, Blank[]], Pattern[$CellContext`c1, Blank[]], Pattern[$CellContext`d1, Blank[]]}, { Pattern[$CellContext`a2, Blank[]], Pattern[$CellContext`b2, Blank[]], Pattern[$CellContext`c2, Blank[]], Pattern[$CellContext`d2, Blank[]]}, { Pattern[$CellContext`a3, Blank[]], Pattern[$CellContext`b3, Blank[]], Pattern[$CellContext`c3, Blank[]], Pattern[$CellContext`d3, Blank[]]}}, { Pattern[$CellContext`col1, Blank[]], Pattern[$CellContext`col2, Blank[]], Pattern[$CellContext`col3, Blank[]]}, Optional[ Pattern[$CellContext`size, Blank[]], 12]] := TableForm[{{ Text[ Style["\[FilledSquare]", 13, $CellContext`col1]], $CellContext`equationFormatter[$CellContext`a1, $CellContext`b1, \ $CellContext`c1, $CellContext`d1, $CellContext`size]}, { Text[ Style["\[FilledSquare]", 13, $CellContext`col2]], $CellContext`equationFormatter[$CellContext`a2, $CellContext`b2, \ $CellContext`c2, $CellContext`d2, $CellContext`size]}, { Text[ Style["\[FilledSquare]", 13, $CellContext`col3]], $CellContext`equationFormatter[$CellContext`a3, $CellContext`b3, \ $CellContext`c3, $CellContext`d3, $CellContext`size]}}], \ $CellContext`equationFormatter[ Pattern[$CellContext`aa, Blank[]], Pattern[$CellContext`bb, Blank[]], Pattern[$CellContext`cc, Blank[]], Pattern[$CellContext`dd, Blank[]], Optional[ Pattern[$CellContext`size, Blank[]], 12]] := TableForm[ Map[Style[#, $CellContext`size, Black]& , { If[$CellContext`aa >= 0, " ", ""], Row[{ ToString[$CellContext`aa, TraditionalForm], Style["x", Italic]}], If[$CellContext`bb >= 0, "+", "-"], Row[{ ToString[ Abs[$CellContext`bb], TraditionalForm], Style["y", Italic]}], If[$CellContext`cc >= 0, "+", "-"], Row[{ ToString[ Abs[$CellContext`cc], TraditionalForm], Style["z", Italic]}], " = ", ToString[ TraditionalForm[$CellContext`dd]]}], TableSpacing -> {1, 5}, TableDirections -> Row], $CellContext`columnVectors[{ Pattern[$CellContext`a1, Blank[]], Pattern[$CellContext`a2, Blank[]], Pattern[$CellContext`a3, Blank[]]}, { Pattern[$CellContext`b1, Blank[]], Pattern[$CellContext`b2, Blank[]], Pattern[$CellContext`b3, Blank[]]}, { Pattern[$CellContext`c1, Blank[]], Pattern[$CellContext`c2, Blank[]], Pattern[$CellContext`c3, Blank[]]}, { Pattern[$CellContext`d1, Blank[]], Pattern[$CellContext`d2, Blank[]], Pattern[$CellContext`d3, Blank[]]}, Pattern[$CellContext`row1Color, Blank[]], Pattern[$CellContext`row2Color, Blank[]], Pattern[$CellContext`row3Color, Blank[]], Pattern[$CellContext`nomRange, Blank[]], Pattern[$CellContext`view\[Theta], Blank[]], Pattern[$CellContext`view\[Phi], Blank[]], Pattern[$CellContext`solFlag, Blank[]], Pattern[$CellContext`solx, Blank[]], Pattern[$CellContext`soly, Blank[]], Pattern[$CellContext`solz, Blank[]], Pattern[$CellContext`zoom, Blank[]]] := Module[{$CellContext`vec1Pic, $CellContext`vec2Pic, \ $CellContext`vec3Pic, $CellContext`vec4Pic, $CellContext`axesPic, \ $CellContext`specialPic, $CellContext`colPlaneParams, $CellContext`matRank, \ $CellContext`range}, $CellContext`range = $CellContext`nomRange \ $CellContext`zoom; $CellContext`matRank = MatrixRank[{{$CellContext`a1, $CellContext`b1, $CellContext`c1}, \ {$CellContext`a2, $CellContext`b2, $CellContext`c2}, {$CellContext`a3, \ $CellContext`b3, $CellContext`c3}}]; $CellContext`specialPic = Graphics3D[ Point[{0, 0, 0}]]; If[$CellContext`matRank == 2, $CellContext`colPlaneParams = \ $CellContext`rank2PlaneThroughColsOfA[{$CellContext`a1, $CellContext`a2, \ $CellContext`a3}, {$CellContext`b1, $CellContext`b2, $CellContext`b3}, \ {$CellContext`c1, $CellContext`c2, $CellContext`c3}]; $CellContext`specialPic = \ $CellContext`plotPlane[$CellContext`colPlaneParams, $CellContext`range, Brown]; Null]; If[$CellContext`matRank == 1, $CellContext`scaler = 0.9 ($CellContext`range/(1 + Min[{$CellContext`a1, $CellContext`a2, $CellContext`a3}])); \ $CellContext`specialPic = Graphics3D[{ Thickness[0.015], Dashing[{0.01, 0.03}], Brown, Line[{{(-$CellContext`scaler) $CellContext`a1, \ (-$CellContext`scaler) $CellContext`a2, (-$CellContext`scaler) \ $CellContext`a3}, {$CellContext`scaler $CellContext`a1, $CellContext`scaler \ $CellContext`a2, $CellContext`scaler $CellContext`a3}}]}]; Null]; $CellContext`colPlaneParams = \ $CellContext`rank2PlaneThroughColsOfA[{$CellContext`a1, $CellContext`a2, \ $CellContext`a3}, {$CellContext`b1, $CellContext`b2, $CellContext`b3}, \ {$CellContext`c1, $CellContext`c2, $CellContext`c3}]; $CellContext`vec1Pic = \ $CellContext`makeVectorPic[{$CellContext`a1, $CellContext`a2, \ $CellContext`a3}, Green]; $CellContext`vec2Pic = \ $CellContext`makeVectorPic[{$CellContext`b1, $CellContext`b2, \ $CellContext`b3}, Orange]; $CellContext`vec3Pic = \ $CellContext`makeVectorPic[{$CellContext`c1, $CellContext`c2, \ $CellContext`c3}, Blue]; $CellContext`vec4Pic = \ $CellContext`makeVectorPic[{$CellContext`d1, $CellContext`d2, \ $CellContext`d3}, Red]; $CellContext`axesPic = \ $CellContext`make3Daxes[$CellContext`range, $CellContext`row1Color, \ $CellContext`row2Color, $CellContext`row3Color, "row 1", "row 2", "row 3", False]; Show[$CellContext`axesPic, $CellContext`vec1Pic, \ $CellContext`vec2Pic, $CellContext`vec3Pic, $CellContext`vec4Pic, \ $CellContext`specialPic, Boxed -> False, ViewPoint -> {($CellContext`range Cos[$CellContext`view\[Theta]]) Cos[$CellContext`view\[Phi]], ($CellContext`range Sin[$CellContext`view\[Theta]]) Cos[$CellContext`view\[Phi]], $CellContext`range Sin[$CellContext`view\[Phi]]}, ViewVertical -> {0, 0, 1}, ImageSize -> {500, 250}, SphericalRegion -> True, Lighting -> "Neutral", PlotRange -> {{(-1.1) $CellContext`range, 1.1 $CellContext`range}, {(-1.2) $CellContext`range, 1.2 $CellContext`range}, {(-1.2) $CellContext`range, 1.2 $CellContext`range}}]], \ $CellContext`rank2PlaneThroughColsOfA[{ Pattern[$CellContext`a1, Blank[]], Pattern[$CellContext`a2, Blank[]], Pattern[$CellContext`a3, Blank[]]}, { Pattern[$CellContext`b1, Blank[]], Pattern[$CellContext`b2, Blank[]], Pattern[$CellContext`b3, Blank[]]}, { Pattern[$CellContext`c1, Blank[]], Pattern[$CellContext`c2, Blank[]], Pattern[$CellContext`c3, Blank[]]}] := Module[{$CellContext`v1, $CellContext`v2, $CellContext`v3, \ $CellContext`mat}, $CellContext`mat = {{$CellContext`a1, $CellContext`b1, \ $CellContext`c1}, {$CellContext`a2, $CellContext`b2, $CellContext`c2}, \ {$CellContext`a3, $CellContext`b3, $CellContext`c3}}; If[MatrixRank[$CellContext`mat] == 2, If[ And[ Norm[{$CellContext`a1, $CellContext`a2, $CellContext`a3}] != 0, Norm[{$CellContext`b1, $CellContext`b2, $CellContext`b3}] != 0], {$CellContext`v1, $CellContext`v2, $CellContext`v3} = \ $CellContext`cprod[{$CellContext`a1, $CellContext`a2, $CellContext`a3}, \ {$CellContext`b1, $CellContext`b2, $CellContext`b3}]; Null, If[ And[ Norm[{$CellContext`a1, $CellContext`a2, $CellContext`a3}] != 0, Norm[{$CellContext`c1, $CellContext`c2, $CellContext`c3}] != 0], {$CellContext`v1, $CellContext`v2, $CellContext`v3} = \ $CellContext`cprod[{$CellContext`a1, $CellContext`a2, $CellContext`a3}, \ {$CellContext`c1, $CellContext`c2, $CellContext`c3}]; Null, {$CellContext`v1, $CellContext`v2, $CellContext`v3} = \ $CellContext`cprod[{$CellContext`b1, $CellContext`b2, $CellContext`b3}, \ {$CellContext`c1, $CellContext`c2, $CellContext`c3}]]; Null]; Null, {$CellContext`v1, $CellContext`v2, $CellContext`v3} = {0, 0, 0}; Null]; {$CellContext`v1, $CellContext`v2, $CellContext`v3, 0}], $CellContext`cprod[{ Pattern[$CellContext`a1, Blank[]], Pattern[$CellContext`a2, Blank[]], Pattern[$CellContext`a3, Blank[]]}, { Pattern[$CellContext`b1, Blank[]], Pattern[$CellContext`b2, Blank[]], Pattern[$CellContext`b3, Blank[]]}] := {(-$CellContext`a3) $CellContext`b2 + \ $CellContext`a2 $CellContext`b3, $CellContext`a3 $CellContext`b1 - \ $CellContext`a1 $CellContext`b3, (-$CellContext`a2) $CellContext`b1 + \ $CellContext`a1 $CellContext`b2}, $CellContext`plotPlane[{ Pattern[$CellContext`a, Blank[]], Pattern[$CellContext`b, Blank[]], Pattern[$CellContext`c, Blank[]], Pattern[$CellContext`d, Blank[]]}, Pattern[$CellContext`range, Blank[]], Pattern[$CellContext`color, Blank[]]] := Module[{$CellContext`x0, $CellContext`y0, $CellContext`z0, \ $CellContext`x1, $CellContext`y1, $CellContext`z1, $CellContext`x2, \ $CellContext`y2, $CellContext`z2, $CellContext`x3, $CellContext`y3, \ $CellContext`z3, $CellContext`myRange}, $CellContext`myRange = \ $CellContext`range 1.7; If[$CellContext`c != 0, $CellContext`x0 = -$CellContext`myRange; $CellContext`y0 = \ -$CellContext`myRange; $CellContext`z0 = ($CellContext`d - $CellContext`a \ $CellContext`x0 - $CellContext`b $CellContext`y0)/$CellContext`c; \ $CellContext`x1 = -$CellContext`myRange; $CellContext`y1 = \ $CellContext`myRange; $CellContext`z1 = ($CellContext`d - $CellContext`a \ $CellContext`x1 - $CellContext`b $CellContext`y1)/$CellContext`c; \ $CellContext`x2 = $CellContext`myRange; $CellContext`y2 = \ -$CellContext`myRange; $CellContext`z2 = ($CellContext`d - $CellContext`a \ $CellContext`x2 - $CellContext`b $CellContext`y2)/$CellContext`c; \ $CellContext`x3 = $CellContext`myRange; $CellContext`y3 = \ $CellContext`myRange; $CellContext`z3 = ($CellContext`d - $CellContext`a \ $CellContext`x3 - $CellContext`b $CellContext`y3)/$CellContext`c; Null, If[$CellContext`b != 0, $CellContext`x0 = -$CellContext`myRange; $CellContext`y0 = \ ($CellContext`d - $CellContext`a $CellContext`x0)/$CellContext`b; \ $CellContext`z0 = -$CellContext`myRange; $CellContext`x1 = \ -$CellContext`myRange; $CellContext`y1 = $CellContext`y0; $CellContext`z1 = \ $CellContext`myRange; $CellContext`x2 = $CellContext`myRange; $CellContext`y2 = \ ($CellContext`d - $CellContext`a $CellContext`x2)/$CellContext`b; \ $CellContext`z2 = -$CellContext`myRange; $CellContext`x3 = \ $CellContext`myRange; $CellContext`y3 = $CellContext`y2; $CellContext`z3 = \ $CellContext`myRange; Null, If[$CellContext`a != 0, $CellContext`x0 = $CellContext`d/$CellContext`a; \ $CellContext`y0 = -$CellContext`myRange; $CellContext`z0 = \ -$CellContext`myRange; $CellContext`x1 = $CellContext`d/$CellContext`a; \ $CellContext`y1 = $CellContext`myRange; $CellContext`z1 = \ -$CellContext`myRange; $CellContext`x2 = $CellContext`d/$CellContext`a; \ $CellContext`y2 = -$CellContext`myRange; $CellContext`z2 = \ $CellContext`myRange; $CellContext`x3 = $CellContext`d/$CellContext`a; \ $CellContext`y3 = $CellContext`myRange; $CellContext`z3 = \ $CellContext`myRange; Null, $CellContext`x0 = 0; $CellContext`y0 = 0; $CellContext`z0 = 0; $CellContext`x1 = 0; $CellContext`y1 = 0; $CellContext`z1 = 0; $CellContext`x2 = 0; $CellContext`y2 = 0; $CellContext`z2 = 0; $CellContext`x3 = 0; $CellContext`y3 = 0; $CellContext`z3 = 0; Null]]]; Graphics3D[{$CellContext`color, Polygon[{{$CellContext`x0, $CellContext`y0, $CellContext`z0}, \ {$CellContext`x1, $CellContext`y1, $CellContext`z1}, {$CellContext`x3, \ $CellContext`y3, $CellContext`z3}, {$CellContext`x2, $CellContext`y2, \ $CellContext`z2}}]}]], $CellContext`scaler = -2.4, $CellContext`makeVectorPic[{ Pattern[$CellContext`a, Blank[]], Pattern[$CellContext`b, Blank[]], Pattern[$CellContext`c, Blank[]]}, Pattern[$CellContext`color, Blank[]]] := Module[{$CellContext`gr1}, $CellContext`gr1 = Graphics3D[{$CellContext`color, Arrowheads[Large], Arrow[ Tube[{{0, 0, 0}, {$CellContext`a, $CellContext`b, $CellContext`c}}, 0.08]]}]; Show[$CellContext`gr1]], $CellContext`make3Daxes[ Pattern[$CellContext`range, Blank[]], Pattern[$CellContext`xColor, Blank[]], Pattern[$CellContext`yColor, Blank[]], Pattern[$CellContext`zColor, Blank[]], Pattern[$CellContext`labx, Blank[]], Pattern[$CellContext`laby, Blank[]], Pattern[$CellContext`labz, Blank[]], Optional[ Pattern[$CellContext`simpleAxesQ, Blank[]], False]] := Module[{$CellContext`axGrid, $CellContext`step, \ $CellContext`simpleAxes}, $CellContext`step = $CellContext`range/ 3; $CellContext`axGrid = \ $CellContext`axesGrid[$CellContext`range, $CellContext`step]; If[$CellContext`simpleAxesQ, $CellContext`simpleAxes = Graphics3D[ Style[{ Line[{{0, 0, -$CellContext`range}, { 0, 0, $CellContext`range}}], Line[{{-$CellContext`range, 0, 0}, {$CellContext`range, 0, 0}}], Line[{{0, -$CellContext`range, 0}, { 0, $CellContext`range, 0}}]}, Black, Thick]]; Null, $CellContext`simpleAxes = Graphics3D[ Point[{0, 0, 0}]]; Null]; Show[$CellContext`axGrid, $CellContext`simpleAxes, Graphics3D[ Text[ Style[$CellContext`labx, Italic, 16], {$CellContext`range 1.2, 0, 0}, Background -> Lighter[$CellContext`xColor, 0.7]]], Graphics3D[ Text[ Style[$CellContext`laby, Italic, 0.8, 16], { 0, $CellContext`range 1.2, 0}, Background -> Lighter[$CellContext`yColor, 0.7]]], Graphics3D[ Text[ Style[$CellContext`labz, Italic, 16], { 0, 0, $CellContext`range 1.2}, Background -> Lighter[$CellContext`zColor, 0.7]]], Boxed -> False, Axes -> Not[$CellContext`simpleAxesQ], BaseStyle -> 12, AxesOrigin -> {0, 0, 0}, AxesStyle -> Directive[Black]]], $CellContext`axesGrid[ Pattern[$CellContext`range, Blank[]], Optional[ Pattern[$CellContext`step, Blank[]], 1]] := Graphics3D[{Thin, GrayLevel[0.8], Table[ Line[{{-$CellContext`range, $CellContext`y, 0}, {$CellContext`range, $CellContext`y, 0}}], {$CellContext`y, -$CellContext`range, \ $CellContext`range, $CellContext`step}], Table[ Line[{{$CellContext`x, -$CellContext`range, 0}, {$CellContext`x, $CellContext`range, 0}}], {$CellContext`x, -$CellContext`range, \ $CellContext`range, $CellContext`step}], Table[ Line[{{$CellContext`x, 0, -$CellContext`range}, {$CellContext`x, 0, $CellContext`range}}], {$CellContext`x, \ -$CellContext`range, $CellContext`range, $CellContext`step}], Table[ Line[{{-$CellContext`range, 0, $CellContext`z}, {$CellContext`range, 0, $CellContext`z}}], {$CellContext`z, -$CellContext`range, \ $CellContext`range, $CellContext`step}], Table[ Line[{{0, $CellContext`y, -$CellContext`range}, { 0, $CellContext`y, $CellContext`range}}], {$CellContext`y, \ -$CellContext`range, $CellContext`range, $CellContext`step}], Table[ Line[{{0, -$CellContext`range, $CellContext`z}, { 0, $CellContext`range, $CellContext`z}}], {$CellContext`z, \ -$CellContext`range, $CellContext`range, $CellContext`step}]}], Attributes[PlotRange] = {ReadProtected}, $CellContext`rowView[{ Pattern[$CellContext`solx, Blank[]], Pattern[$CellContext`soly, Blank[]], Pattern[$CellContext`solz, Blank[]]}, Pattern[$CellContext`solflag, Blank[]], {{ Pattern[$CellContext`a1, Blank[]], Pattern[$CellContext`b1, Blank[]], Pattern[$CellContext`c1, Blank[]], Pattern[$CellContext`d1, Blank[]]}, { Pattern[$CellContext`a2, Blank[]], Pattern[$CellContext`b2, Blank[]], Pattern[$CellContext`c2, Blank[]], Pattern[$CellContext`d2, Blank[]]}, { Pattern[$CellContext`a3, Blank[]], Pattern[$CellContext`b3, Blank[]], Pattern[$CellContext`c3, Blank[]], Pattern[$CellContext`d3, Blank[]]}}, Pattern[$CellContext`row1Color, Blank[]], Pattern[$CellContext`row2Color, Blank[]], Pattern[$CellContext`row3Color, Blank[]], Pattern[$CellContext`nomRange, Blank[]], Pattern[$CellContext`view\[Theta], Blank[]], Pattern[$CellContext`view\[Phi], Blank[]], Pattern[$CellContext`zoom, Blank[]]] := Module[{$CellContext`solPic, $CellContext`axesPic, \ $CellContext`surface1Pic, $CellContext`surface2Pic, $CellContext`surface3Pic, \ $CellContext`range}, $CellContext`range = $CellContext`nomRange \ $CellContext`zoom; Switch[$CellContext`solflag, 0, $CellContext`solPic = Graphics3D[{ Point[{0, 0, 0}]}]; Null, 1, $CellContext`solPic = $CellContext`plotPlane[{$CellContext`a1, \ $CellContext`b1, $CellContext`c1, $CellContext`d1}, $CellContext`range, Red]; Null, 2, $CellContext`solPic = \ $CellContext`plotIntersectionOf2Planes[{$CellContext`solx, $CellContext`soly, \ $CellContext`solz}, $CellContext`range, Red]; Null, 3, $CellContext`solPic = Graphics3D[{Red, Sphere[{$CellContext`solx, $CellContext`soly, \ $CellContext`solz}, $CellContext`range/16]}]; Null, 4, $CellContext`solPic = Graphics3D[{ Point[{0, 0, 0}]}]; Null]; $CellContext`axesPic = \ $CellContext`make3Daxes[$CellContext`range, White, White, White, "x", "y", "z", True]; $CellContext`surface1Pic = \ $CellContext`plotPlane[{$CellContext`a1, $CellContext`b1, $CellContext`c1, \ $CellContext`d1}, $CellContext`range, $CellContext`row1Color]; \ $CellContext`surface2Pic = $CellContext`plotPlane[{$CellContext`a2, \ $CellContext`b2, $CellContext`c2, $CellContext`d2}, $CellContext`range, \ $CellContext`row2Color]; $CellContext`surface3Pic = \ $CellContext`plotPlane[{$CellContext`a3, $CellContext`b3, $CellContext`c3, \ $CellContext`d3}, $CellContext`range, $CellContext`row3Color]; Show[$CellContext`axesPic, $CellContext`surface1Pic, \ $CellContext`surface2Pic, $CellContext`surface3Pic, $CellContext`solPic, Boxed -> False, ViewPoint -> {($CellContext`range Cos[$CellContext`view\[Theta]]) Cos[$CellContext`view\[Phi]], ($CellContext`range Sin[$CellContext`view\[Theta]]) Cos[$CellContext`view\[Phi]], $CellContext`range Sin[$CellContext`view\[Phi]]}, ViewVertical -> {0, 0, 1}, ImageSize -> {500, 250}, SphericalRegion -> True, Lighting -> "Neutral", PlotRange -> {{(-1.1) $CellContext`range, 1.1 $CellContext`range}, {(-1.2) $CellContext`range, 1.2 $CellContext`range}, {(-1.2) $CellContext`range, 1.2 $CellContext`range}}]], \ $CellContext`plotIntersectionOf2Planes[{ Pattern[$CellContext`solx, Blank[]], Pattern[$CellContext`soly, Blank[]], Pattern[$CellContext`solz, Blank[]]}, Pattern[$CellContext`range, Blank[]], Pattern[$CellContext`color, Blank[]]] := Module[{$CellContext`varName}, $CellContext`varName = \ $CellContext`getVarFromSols[{$CellContext`solx, $CellContext`soly, \ $CellContext`solz}]; $CellContext`plotLineFromSolns[$CellContext`varName, \ {$CellContext`solx, $CellContext`soly, $CellContext`solz}, \ $CellContext`range, $CellContext`color]], $CellContext`getVarFromSols[{ Pattern[$CellContext`solx, Blank[]], Pattern[$CellContext`soly, Blank[]], Pattern[$CellContext`solz, Blank[]]}] := First[ Variables[{$CellContext`solx, $CellContext`soly, \ $CellContext`solz}]], $CellContext`plotLineFromSolns[ Pattern[$CellContext`pVar, Blank[]], { Pattern[$CellContext`solx, Blank[]], Pattern[$CellContext`soly, Blank[]], Pattern[$CellContext`solz, Blank[]]}, Pattern[$CellContext`range, Blank[]], Pattern[$CellContext`color, Blank[]]] := ParametricPlot3D[{$CellContext`solx, $CellContext`soly, \ $CellContext`solz}, {$CellContext`pVar, -$CellContext`range, \ $CellContext`range}, PlotStyle -> { Thickness[$CellContext`range/800], Red}], $CellContext`getExamples[ Pattern[$CellContext`exType, Blank[]]] := Switch[$CellContext`exType, 1, { RandomInteger[{-10, 10}, 4], RandomInteger[{-10, 10}, 4], RandomInteger[{-10, 10}, 4]}, 2, {{1, 2, 3, 6}, {-2, -2, 2, 2}, { 10, -8, 2, 4}}, 3, {{2, 1, 3, 4}, {2, 7, 5, 3}, {4, 8, 8, 7}}, 4, {{-4, -2, -3, 8}, {-4, -2, -3, 8}, {-4, -2, -3, 8}}, 5, {{0, 0, -3, 7}, {7, 0, -4, 0}, {7, 0, 0, 10}}, 6, {{1, 2, 3, 4}, {4, 3, 2, 1}, {-1, -2, -1, 6}}]}; Typeset`initDone$$ = True); ReleaseHold[ HoldComplete[{$CellContext`plotPlane[{ Pattern[$CellContext`a, Blank[]], Pattern[$CellContext`b, Blank[]], Pattern[$CellContext`c, Blank[]], Pattern[$CellContext`d, Blank[]]}, Pattern[$CellContext`range, Blank[]], Pattern[$CellContext`color, Blank[]]] := Module[{$CellContext`x0, $CellContext`y0, $CellContext`z0, \ $CellContext`x1, $CellContext`y1, $CellContext`z1, $CellContext`x2, \ $CellContext`y2, $CellContext`z2, $CellContext`x3, $CellContext`y3, \ $CellContext`z3, $CellContext`myRange}, $CellContext`myRange = \ $CellContext`range 1.7; If[$CellContext`c != 0, $CellContext`x0 = -$CellContext`myRange; $CellContext`y0 = \ -$CellContext`myRange; $CellContext`z0 = ($CellContext`d - $CellContext`a \ $CellContext`x0 - $CellContext`b $CellContext`y0)/$CellContext`c; \ $CellContext`x1 = -$CellContext`myRange; $CellContext`y1 = \ $CellContext`myRange; $CellContext`z1 = ($CellContext`d - $CellContext`a \ $CellContext`x1 - $CellContext`b $CellContext`y1)/$CellContext`c; \ $CellContext`x2 = $CellContext`myRange; $CellContext`y2 = \ -$CellContext`myRange; $CellContext`z2 = ($CellContext`d - $CellContext`a \ $CellContext`x2 - $CellContext`b $CellContext`y2)/$CellContext`c; \ $CellContext`x3 = $CellContext`myRange; $CellContext`y3 = \ $CellContext`myRange; $CellContext`z3 = ($CellContext`d - $CellContext`a \ $CellContext`x3 - $CellContext`b $CellContext`y3)/$CellContext`c; Null, If[$CellContext`b != 0, $CellContext`x0 = -$CellContext`myRange; $CellContext`y0 = \ ($CellContext`d - $CellContext`a $CellContext`x0)/$CellContext`b; \ $CellContext`z0 = -$CellContext`myRange; $CellContext`x1 = \ -$CellContext`myRange; $CellContext`y1 = $CellContext`y0; $CellContext`z1 = \ $CellContext`myRange; $CellContext`x2 = $CellContext`myRange; $CellContext`y2 = \ ($CellContext`d - $CellContext`a $CellContext`x2)/$CellContext`b; \ $CellContext`z2 = -$CellContext`myRange; $CellContext`x3 = \ $CellContext`myRange; $CellContext`y3 = $CellContext`y2; $CellContext`z3 = \ $CellContext`myRange; Null, If[$CellContext`a != 0, $CellContext`x0 = $CellContext`d/$CellContext`a; \ $CellContext`y0 = -$CellContext`myRange; $CellContext`z0 = \ -$CellContext`myRange; $CellContext`x1 = $CellContext`d/$CellContext`a; \ $CellContext`y1 = $CellContext`myRange; $CellContext`z1 = \ -$CellContext`myRange; $CellContext`x2 = $CellContext`d/$CellContext`a; \ $CellContext`y2 = -$CellContext`myRange; $CellContext`z2 = \ $CellContext`myRange; $CellContext`x3 = $CellContext`d/$CellContext`a; \ $CellContext`y3 = $CellContext`myRange; $CellContext`z3 = \ $CellContext`myRange; Null, $CellContext`x0 = 0; $CellContext`y0 = 0; $CellContext`z0 = 0; $CellContext`x1 = 0; $CellContext`y1 = 0; $CellContext`z1 = 0; $CellContext`x2 = 0; $CellContext`y2 = 0; $CellContext`z2 = 0; $CellContext`x3 = 0; $CellContext`y3 = 0; $CellContext`z3 = 0; Null]]]; Graphics3D[{$CellContext`color, Polygon[{{$CellContext`x0, $CellContext`y0, $CellContext`z0}, \ {$CellContext`x1, $CellContext`y1, $CellContext`z1}, {$CellContext`x3, \ $CellContext`y3, $CellContext`z3}, {$CellContext`x2, $CellContext`y2, \ $CellContext`z2}}]}]]; Null, $CellContext`getVarFromSols[{ Pattern[$CellContext`solx, Blank[]], Pattern[$CellContext`soly, Blank[]], Pattern[$CellContext`solz, Blank[]]}] := First[ Variables[{$CellContext`solx, $CellContext`soly, \ $CellContext`solz}]]; Null, $CellContext`plotIntersectionOf2Planes[{ Pattern[$CellContext`solx, Blank[]], Pattern[$CellContext`soly, Blank[]], Pattern[$CellContext`solz, Blank[]]}, Pattern[$CellContext`range, Blank[]], Pattern[$CellContext`color, Blank[]]] := Module[{$CellContext`varName}, $CellContext`varName = \ $CellContext`getVarFromSols[{$CellContext`solx, $CellContext`soly, \ $CellContext`solz}]; $CellContext`plotLineFromSolns[$CellContext`varName, \ {$CellContext`solx, $CellContext`soly, $CellContext`solz}, \ $CellContext`range, $CellContext`color]]; Null, $CellContext`plotLineFromSolns[ Pattern[$CellContext`pVar, Blank[]], { Pattern[$CellContext`solx, Blank[]], Pattern[$CellContext`soly, Blank[]], Pattern[$CellContext`solz, Blank[]]}, Pattern[$CellContext`range, Blank[]], Pattern[$CellContext`color, Blank[]]] := ParametricPlot3D[{$CellContext`solx, $CellContext`soly, \ $CellContext`solz}, {$CellContext`pVar, -$CellContext`range, \ $CellContext`range}, PlotStyle -> { Thickness[$CellContext`range/800], Red}]; Null, $CellContext`equationFormatter[ Pattern[$CellContext`aa, Blank[]], Pattern[$CellContext`bb, Blank[]], Pattern[$CellContext`cc, Blank[]], Pattern[$CellContext`dd, Blank[]], Optional[ Pattern[$CellContext`size, Blank[]], 12]] := TableForm[ Map[Style[#, $CellContext`size, Black]& , { If[$CellContext`aa >= 0, " ", ""], Row[{ ToString[$CellContext`aa, TraditionalForm], Style["x", Italic]}], If[$CellContext`bb >= 0, "+", "-"], Row[{ ToString[ Abs[$CellContext`bb], TraditionalForm], Style["y", Italic]}], If[$CellContext`cc >= 0, "+", "-"], Row[{ ToString[ Abs[$CellContext`cc], TraditionalForm], Style["z", Italic]}], " = ", ToString[ TraditionalForm[$CellContext`dd]]}], TableSpacing -> {1, 5}, TableDirections -> Row]; Null, $CellContext`sysFormatter[{{ Pattern[$CellContext`a1, Blank[]], Pattern[$CellContext`b1, Blank[]], Pattern[$CellContext`c1, Blank[]], Pattern[$CellContext`d1, Blank[]]}, { Pattern[$CellContext`a2, Blank[]], Pattern[$CellContext`b2, Blank[]], Pattern[$CellContext`c2, Blank[]], Pattern[$CellContext`d2, Blank[]]}, { Pattern[$CellContext`a3, Blank[]], Pattern[$CellContext`b3, Blank[]], Pattern[$CellContext`c3, Blank[]], Pattern[$CellContext`d3, Blank[]]}}, Optional[ Pattern[$CellContext`size, Blank[]], 12]] := TableForm[{ $CellContext`equationFormatter[$CellContext`a1, $CellContext`b1, \ $CellContext`c1, $CellContext`d1, $CellContext`size], $CellContext`equationFormatter[$CellContext`a2, $CellContext`b2, \ $CellContext`c2, $CellContext`d2, $CellContext`size], $CellContext`equationFormatter[$CellContext`a3, $CellContext`b3, \ $CellContext`c3, $CellContext`d3, $CellContext`size]}]; Null, $CellContext`sysFormatterColor[{{ Pattern[$CellContext`a1, Blank[]], Pattern[$CellContext`b1, Blank[]], Pattern[$CellContext`c1, Blank[]], Pattern[$CellContext`d1, Blank[]]}, { Pattern[$CellContext`a2, Blank[]], Pattern[$CellContext`b2, Blank[]], Pattern[$CellContext`c2, Blank[]], Pattern[$CellContext`d2, Blank[]]}, { Pattern[$CellContext`a3, Blank[]], Pattern[$CellContext`b3, Blank[]], Pattern[$CellContext`c3, Blank[]], Pattern[$CellContext`d3, Blank[]]}}, { Pattern[$CellContext`col1, Blank[]], Pattern[$CellContext`col2, Blank[]], Pattern[$CellContext`col3, Blank[]]}, Optional[ Pattern[$CellContext`size, Blank[]], 12]] := TableForm[{{ Text[ Style["\[FilledSquare]", 13, $CellContext`col1]], $CellContext`equationFormatter[$CellContext`a1, \ $CellContext`b1, $CellContext`c1, $CellContext`d1, $CellContext`size]}, { Text[ Style["\[FilledSquare]", 13, $CellContext`col2]], $CellContext`equationFormatter[$CellContext`a2, \ $CellContext`b2, $CellContext`c2, $CellContext`d2, $CellContext`size]}, { Text[ Style["\[FilledSquare]", 13, $CellContext`col3]], $CellContext`equationFormatter[$CellContext`a3, \ $CellContext`b3, $CellContext`c3, $CellContext`d3, $CellContext`size]}}]; Null, $CellContext`ColInterpFormatter[{{ Pattern[$CellContext`a1, Blank[]], Pattern[$CellContext`b1, Blank[]], Pattern[$CellContext`c1, Blank[]], Pattern[$CellContext`d1, Blank[]]}, { Pattern[$CellContext`a2, Blank[]], Pattern[$CellContext`b2, Blank[]], Pattern[$CellContext`c2, Blank[]], Pattern[$CellContext`d2, Blank[]]}, { Pattern[$CellContext`a3, Blank[]], Pattern[$CellContext`b3, Blank[]], Pattern[$CellContext`c3, Blank[]], Pattern[$CellContext`d3, Blank[]]}}, Optional[ Pattern[$CellContext`size, Blank[]], 12]] := Row[ Map[TraditionalForm, { Row[{ Style["x", Italic], " \[CenterDot]"}], MatrixForm[{{ Text[ Style[$CellContext`a1, $CellContext`size, Green, Bold]]}, { Text[ Style[$CellContext`a2, $CellContext`size, Green, Bold]]}, { Text[ Style[$CellContext`a3, $CellContext`size, Green, Bold]]}}], Row[{" + ", Style["y", Italic], " \[CenterDot]"}], MatrixForm[{{ Text[ Style[$CellContext`b1, $CellContext`size, Orange, Bold]]}, { Text[ Style[$CellContext`b2, $CellContext`size, Orange, Bold]]}, { Text[ Style[$CellContext`b3, $CellContext`size, Orange, Bold]]}}], Row[{" + ", Style["z", Italic], " \[CenterDot]"}], MatrixForm[{{ Text[ Style[$CellContext`c1, $CellContext`size, Blue, Bold]]}, { Text[ Style[$CellContext`c2, $CellContext`size, Blue, Bold]]}, { Text[ Style[$CellContext`c3, $CellContext`size, Blue, Bold]]}}], " = ", MatrixForm[{{ Text[ Style[$CellContext`d1, $CellContext`size, Red, Bold]]}, { Text[ Style[$CellContext`d2, $CellContext`size, Red, Bold]]}, { Text[ Style[$CellContext`d3, $CellContext`size, Red, Bold]]}}]}]], $CellContext`solFormatter[{ Pattern[$CellContext`xsol, Blank[]], Pattern[$CellContext`ysol, Blank[]], Pattern[$CellContext`zsol, Blank[]]}, Optional[ Pattern[$CellContext`size, Blank[]], 12]] := TableForm[ Map[Style[#, $CellContext`size, Black]& , { If[ToString[$CellContext`xsol] == "varX", " ", Row[{ Style["x", Italic], " = ", StringReplace[ ToString[ TraditionalForm[$CellContext`xsol]], { "varY" -> "y", "varZ" -> "z"}]}]], " ", If[ToString[$CellContext`ysol] == "varY", " ", Row[{ Style["y", Italic], " = ", StringReplace[ ToString[ TraditionalForm[$CellContext`ysol]], { "varX" -> "x", "varZ" -> "z"}]}]], " ", If[ToString[$CellContext`zsol] == "varZ", " ", Row[{ Style["z", Italic], " = ", StringReplace[ ToString[ TraditionalForm[$CellContext`zsol]], { "varY" -> "y", "varX" -> "x"}]}]]}], TableSpacing -> {0, 0}, TableDirections -> Column]; Null, $CellContext`solutionFormatter[{ Pattern[$CellContext`xsol, Blank[]], Pattern[$CellContext`ysol, Blank[]], Pattern[$CellContext`zsol, Blank[]], Pattern[$CellContext`solflag, Blank[]]}, Optional[ Pattern[$CellContext`size, Blank[]], 12]] := Switch[$CellContext`solflag, 0, Text[ Style["NO SOLUTION", $CellContext`size, Red]], 1, $CellContext`solFormatter[{$CellContext`xsol, $CellContext`ysol, \ $CellContext`zsol}, $CellContext`size], 2, $CellContext`solFormatter[{$CellContext`xsol, $CellContext`ysol, \ $CellContext`zsol}, $CellContext`size], 3, $CellContext`solFormatter[{$CellContext`xsol, $CellContext`ysol, \ $CellContext`zsol}, $CellContext`size], 4, Text[ Style[ Row[{"\[ForAll] {", Style["x", Italic], ",", Style["y", Italic], ",", Style["z", Italic], "} \[Element] \ \!\(\*SuperscriptBox[\(\[DoubleStruckCapitalR]\), \(3\)]\)"}], \ $CellContext`size, Red]]]; Null, $CellContext`SolveSystem[ Pattern[$CellContext`myMat, Blank[List]]] := Module[{$CellContext`solX, $CellContext`solY, $CellContext`solZ, \ $CellContext`tempSol, $CellContext`solFlag}, {$CellContext`tempSol, \ $CellContext`solFlag} = $CellContext`SolveSystem2[$CellContext`myMat]; \ {$CellContext`solX, $CellContext`solY, $CellContext`solZ} = ReplaceAll[{$CellContext`varX, $CellContext`varY, \ $CellContext`varZ}, $CellContext`tempSol]; {$CellContext`solX, \ $CellContext`solY, $CellContext`solZ, $CellContext`solFlag}]; Null, $CellContext`SolveSystem2[ Pattern[$CellContext`myMat, Blank[List]]] := Quiet[ Module[{$CellContext`tempSol, $CellContext`solFlag, \ $CellContext`x, $CellContext`y, $CellContext`z}, If[Norm[$CellContext`myMat] == 0, $CellContext`tempSol = {}; $CellContext`solFlag = 4; Null, $CellContext`tempSol = Part[ Solve[{Part[$CellContext`myMat, 1, 1] $CellContext`varX + Part[$CellContext`myMat, 1, 2] $CellContext`varY + Part[$CellContext`myMat, 1, 3] $CellContext`varZ == Part[$CellContext`myMat, 1, 4], Part[$CellContext`myMat, 2, 1] $CellContext`varX + Part[$CellContext`myMat, 2, 2] $CellContext`varY + Part[$CellContext`myMat, 2, 3] $CellContext`varZ == Part[$CellContext`myMat, 2, 4], Part[$CellContext`myMat, 3, 1] $CellContext`varX + Part[$CellContext`myMat, 3, 2] $CellContext`varY + Part[$CellContext`myMat, 3, 3] $CellContext`varZ == Part[$CellContext`myMat, 3, 4]}, {$CellContext`varX, $CellContext`varY, \ $CellContext`varZ}], 1]; $CellContext`solFlag = 0; $CellContext`solFlag = Length[$CellContext`tempSol]; If[ Not[ StringFreeQ[ ToString[$CellContext`tempSol], "{}"]], $CellContext`solFlag = 0; $CellContext`tempSol = {}]; Null]; {$CellContext`tempSol, $CellContext`solFlag}]]; Null, { HoldComplete[$CellContext`getExamples[ Pattern[$CellContext`exType, Blank[]]] := Switch[$CellContext`exType, 1, { RandomInteger[{-10, 10}, 4], RandomInteger[{-10, 10}, 4], RandomInteger[{-10, 10}, 4]}, 2, {{1, 2, 3, 6}, {-2, -2, 2, 2}, {10, -8, 2, 4}}, 3, {{2, 1, 3, 4}, {2, 7, 5, 3}, {4, 8, 8, 7}}, 4, {{-4, -2, -3, 8}, {-4, -2, -3, 8}, {-4, -2, -3, 8}}, 5, {{0, 0, -3, 7}, {7, 0, -4, 0}, {7, 0, 0, 10}}, 6, {{1, 2, 3, 4}, {4, 3, 2, 1}, {-1, -2, -1, 6}}]], HoldComplete[Null], HoldComplete[Null]}, $CellContext`axesGrid[ Pattern[$CellContext`range, Blank[]], Optional[ Pattern[$CellContext`step, Blank[]], 1]] := Graphics3D[{Thin, GrayLevel[0.8], Table[ Line[{{-$CellContext`range, $CellContext`y, 0}, {$CellContext`range, $CellContext`y, 0}}], {$CellContext`y, -$CellContext`range, \ $CellContext`range, $CellContext`step}], Table[ Line[{{$CellContext`x, -$CellContext`range, 0}, {$CellContext`x, $CellContext`range, 0}}], {$CellContext`x, -$CellContext`range, \ $CellContext`range, $CellContext`step}], Table[ Line[{{$CellContext`x, 0, -$CellContext`range}, {$CellContext`x, 0, $CellContext`range}}], {$CellContext`x, \ -$CellContext`range, $CellContext`range, $CellContext`step}], Table[ Line[{{-$CellContext`range, 0, $CellContext`z}, {$CellContext`range, 0, $CellContext`z}}], {$CellContext`z, -$CellContext`range, \ $CellContext`range, $CellContext`step}], Table[ Line[{{0, $CellContext`y, -$CellContext`range}, { 0, $CellContext`y, $CellContext`range}}], {$CellContext`y, \ -$CellContext`range, $CellContext`range, $CellContext`step}], Table[ Line[{{0, -$CellContext`range, $CellContext`z}, { 0, $CellContext`range, $CellContext`z}}], {$CellContext`z, \ -$CellContext`range, $CellContext`range, $CellContext`step}]}], \ $CellContext`make3Daxes[ Pattern[$CellContext`range, Blank[]], Pattern[$CellContext`xColor, Blank[]], Pattern[$CellContext`yColor, Blank[]], Pattern[$CellContext`zColor, Blank[]], Pattern[$CellContext`labx, Blank[]], Pattern[$CellContext`laby, Blank[]], Pattern[$CellContext`labz, Blank[]], Optional[ Pattern[$CellContext`simpleAxesQ, Blank[]], False]] := Module[{$CellContext`axGrid, $CellContext`step, \ $CellContext`simpleAxes}, $CellContext`step = $CellContext`range/ 3; $CellContext`axGrid = \ $CellContext`axesGrid[$CellContext`range, $CellContext`step]; If[$CellContext`simpleAxesQ, $CellContext`simpleAxes = Graphics3D[ Style[{ Line[{{0, 0, -$CellContext`range}, { 0, 0, $CellContext`range}}], Line[{{-$CellContext`range, 0, 0}, {$CellContext`range, 0, 0}}], Line[{{0, -$CellContext`range, 0}, { 0, $CellContext`range, 0}}]}, Black, Thick]]; Null, $CellContext`simpleAxes = Graphics3D[ Point[{0, 0, 0}]]; Null]; Show[$CellContext`axGrid, $CellContext`simpleAxes, Graphics3D[ Text[ Style[$CellContext`labx, Italic, 16], {$CellContext`range 1.2, 0, 0}, Background -> Lighter[$CellContext`xColor, 0.7]]], Graphics3D[ Text[ Style[$CellContext`laby, Italic, 0.8, 16], { 0, $CellContext`range 1.2, 0}, Background -> Lighter[$CellContext`yColor, 0.7]]], Graphics3D[ Text[ Style[$CellContext`labz, Italic, 16], { 0, 0, $CellContext`range 1.2}, Background -> Lighter[$CellContext`zColor, 0.7]]], Boxed -> False, Axes -> Not[$CellContext`simpleAxesQ], BaseStyle -> 12, AxesOrigin -> {0, 0, 0}, AxesStyle -> Directive[Black]]], { HoldComplete[$CellContext`cprod[{ Pattern[$CellContext`a1, Blank[]], Pattern[$CellContext`a2, Blank[]], Pattern[$CellContext`a3, Blank[]]}, { Pattern[$CellContext`b1, Blank[]], Pattern[$CellContext`b2, Blank[]], Pattern[$CellContext`b3, Blank[]]}] := {(-$CellContext`a3) $CellContext`b2 + \ $CellContext`a2 $CellContext`b3, $CellContext`a3 $CellContext`b1 - \ $CellContext`a1 $CellContext`b3, (-$CellContext`a2) $CellContext`b1 + \ $CellContext`a1 $CellContext`b2}; Null], HoldComplete[Null], HoldComplete[$CellContext`rank2PlaneThroughColsOfA[{ Pattern[$CellContext`a1, Blank[]], Pattern[$CellContext`a2, Blank[]], Pattern[$CellContext`a3, Blank[]]}, { Pattern[$CellContext`b1, Blank[]], Pattern[$CellContext`b2, Blank[]], Pattern[$CellContext`b3, Blank[]]}, { Pattern[$CellContext`c1, Blank[]], Pattern[$CellContext`c2, Blank[]], Pattern[$CellContext`c3, Blank[]]}] := Module[{$CellContext`v1, $CellContext`v2, $CellContext`v3, \ $CellContext`mat}, $CellContext`mat = {{$CellContext`a1, $CellContext`b1, \ $CellContext`c1}, {$CellContext`a2, $CellContext`b2, $CellContext`c2}, \ {$CellContext`a3, $CellContext`b3, $CellContext`c3}}; If[MatrixRank[$CellContext`mat] == 2, If[ And[ Norm[{$CellContext`a1, $CellContext`a2, $CellContext`a3}] != 0, Norm[{$CellContext`b1, $CellContext`b2, \ $CellContext`b3}] != 0], {$CellContext`v1, $CellContext`v2, $CellContext`v3} = \ $CellContext`cprod[{$CellContext`a1, $CellContext`a2, $CellContext`a3}, \ {$CellContext`b1, $CellContext`b2, $CellContext`b3}]; Null, If[ And[ Norm[{$CellContext`a1, $CellContext`a2, $CellContext`a3}] != 0, Norm[{$CellContext`c1, $CellContext`c2, \ $CellContext`c3}] != 0], {$CellContext`v1, $CellContext`v2, $CellContext`v3} = \ $CellContext`cprod[{$CellContext`a1, $CellContext`a2, $CellContext`a3}, \ {$CellContext`c1, $CellContext`c2, $CellContext`c3}]; Null, {$CellContext`v1, $CellContext`v2, $CellContext`v3} = \ $CellContext`cprod[{$CellContext`b1, $CellContext`b2, $CellContext`b3}, \ {$CellContext`c1, $CellContext`c2, $CellContext`c3}]]; Null]; Null, {$CellContext`v1, $CellContext`v2, $CellContext`v3} = {0, 0, 0}; Null]; {$CellContext`v1, $CellContext`v2, $CellContext`v3, 0}]]}, $CellContext`makeVectorPic[{ Pattern[$CellContext`a, Blank[]], Pattern[$CellContext`b, Blank[]], Pattern[$CellContext`c, Blank[]]}, Pattern[$CellContext`color, Blank[]]] := Graphics3D[{$CellContext`color, Arrowheads[Large], Arrow[ Tube[{{0, 0, 0}, {$CellContext`a, $CellContext`b, $CellContext`c}}, 0.08]]}]; Null, $CellContext`columnVectors[{ Pattern[$CellContext`a1, Blank[]], Pattern[$CellContext`a2, Blank[]], Pattern[$CellContext`a3, Blank[]]}, { Pattern[$CellContext`b1, Blank[]], Pattern[$CellContext`b2, Blank[]], Pattern[$CellContext`b3, Blank[]]}, { Pattern[$CellContext`c1, Blank[]], Pattern[$CellContext`c2, Blank[]], Pattern[$CellContext`c3, Blank[]]}, { Pattern[$CellContext`d1, Blank[]], Pattern[$CellContext`d2, Blank[]], Pattern[$CellContext`d3, Blank[]]}, Pattern[$CellContext`row1Color, Blank[]], Pattern[$CellContext`row2Color, Blank[]], Pattern[$CellContext`row3Color, Blank[]], Pattern[$CellContext`nomRange, Blank[]], Pattern[$CellContext`view\[Theta], Blank[]], Pattern[$CellContext`view\[Phi], Blank[]], Pattern[$CellContext`solFlag, Blank[]], Pattern[$CellContext`solx, Blank[]], Pattern[$CellContext`soly, Blank[]], Pattern[$CellContext`solz, Blank[]], Pattern[$CellContext`zoom, Blank[]]] := Module[{$CellContext`vec1Pic, $CellContext`vec2Pic, \ $CellContext`vec3Pic, $CellContext`vec4Pic, $CellContext`axesPic, \ $CellContext`specialPic, $CellContext`colPlaneParams, $CellContext`matRank, \ $CellContext`range}, $CellContext`range = $CellContext`nomRange \ $CellContext`zoom; $CellContext`matRank = MatrixRank[{{$CellContext`a1, $CellContext`b1, $CellContext`c1}, \ {$CellContext`a2, $CellContext`b2, $CellContext`c2}, {$CellContext`a3, \ $CellContext`b3, $CellContext`c3}}]; $CellContext`specialPic = Graphics3D[ Point[{0, 0, 0}]]; If[$CellContext`matRank == 2, $CellContext`colPlaneParams = \ $CellContext`rank2PlaneThroughColsOfA[{$CellContext`a1, $CellContext`a2, \ $CellContext`a3}, {$CellContext`b1, $CellContext`b2, $CellContext`b3}, \ {$CellContext`c1, $CellContext`c2, $CellContext`c3}]; $CellContext`specialPic = \ $CellContext`plotPlane[$CellContext`colPlaneParams, $CellContext`range, Brown]; Null]; If[$CellContext`matRank == 1, $CellContext`scaler = 0.9 ($CellContext`range/(1 + Min[{$CellContext`a1, $CellContext`a2, $CellContext`a3}])); \ $CellContext`specialPic = Graphics3D[{ Thickness[0.015], Dashing[{0.01, 0.03}], Brown, Line[{{(-$CellContext`scaler) $CellContext`a1, \ (-$CellContext`scaler) $CellContext`a2, (-$CellContext`scaler) \ $CellContext`a3}, {$CellContext`scaler $CellContext`a1, $CellContext`scaler \ $CellContext`a2, $CellContext`scaler $CellContext`a3}}]}]; Null]; $CellContext`colPlaneParams = \ $CellContext`rank2PlaneThroughColsOfA[{$CellContext`a1, $CellContext`a2, \ $CellContext`a3}, {$CellContext`b1, $CellContext`b2, $CellContext`b3}, \ {$CellContext`c1, $CellContext`c2, $CellContext`c3}]; $CellContext`vec1Pic = \ $CellContext`makeVectorPic[{$CellContext`a1, $CellContext`a2, \ $CellContext`a3}, Green]; $CellContext`vec2Pic = \ $CellContext`makeVectorPic[{$CellContext`b1, $CellContext`b2, \ $CellContext`b3}, Orange]; $CellContext`vec3Pic = \ $CellContext`makeVectorPic[{$CellContext`c1, $CellContext`c2, \ $CellContext`c3}, Blue]; $CellContext`vec4Pic = \ $CellContext`makeVectorPic[{$CellContext`d1, $CellContext`d2, \ $CellContext`d3}, Red]; $CellContext`axesPic = \ $CellContext`make3Daxes[$CellContext`range, $CellContext`row1Color, \ $CellContext`row2Color, $CellContext`row3Color, "row 1", "row 2", "row 3", False]; Show[$CellContext`axesPic, $CellContext`vec1Pic, \ $CellContext`vec2Pic, $CellContext`vec3Pic, $CellContext`vec4Pic, \ $CellContext`specialPic, Boxed -> False, ViewPoint -> {$CellContext`range Cos[$CellContext`view\[Theta]] Cos[$CellContext`view\[Phi]], $CellContext`range Sin[$CellContext`view\[Theta]] Cos[$CellContext`view\[Phi]], $CellContext`range Sin[$CellContext`view\[Phi]]}, ViewVertical -> {0, 0, 1}, ImageSize -> {500, 250}, SphericalRegion -> True, Lighting -> "Neutral", PlotRange -> {{(-1.1) $CellContext`range, 1.1 $CellContext`range}, {(-1.2) $CellContext`range, 1.2 $CellContext`range}, {(-1.2) $CellContext`range, 1.2 $CellContext`range}}]], $CellContext`rowView[{ Pattern[$CellContext`solx, Blank[]], Pattern[$CellContext`soly, Blank[]], Pattern[$CellContext`solz, Blank[]]}, Pattern[$CellContext`solflag, Blank[]], {{ Pattern[$CellContext`a1, Blank[]], Pattern[$CellContext`b1, Blank[]], Pattern[$CellContext`c1, Blank[]], Pattern[$CellContext`d1, Blank[]]}, { Pattern[$CellContext`a2, Blank[]], Pattern[$CellContext`b2, Blank[]], Pattern[$CellContext`c2, Blank[]], Pattern[$CellContext`d2, Blank[]]}, { Pattern[$CellContext`a3, Blank[]], Pattern[$CellContext`b3, Blank[]], Pattern[$CellContext`c3, Blank[]], Pattern[$CellContext`d3, Blank[]]}}, Pattern[$CellContext`row1Color, Blank[]], Pattern[$CellContext`row2Color, Blank[]], Pattern[$CellContext`row3Color, Blank[]], Pattern[$CellContext`nomRange, Blank[]], Pattern[$CellContext`view\[Theta], Blank[]], Pattern[$CellContext`view\[Phi], Blank[]], Pattern[$CellContext`zoom, Blank[]]] := Module[{$CellContext`solPic, $CellContext`axesPic, \ $CellContext`surface1Pic, $CellContext`surface2Pic, $CellContext`surface3Pic, \ $CellContext`range}, $CellContext`range = $CellContext`nomRange \ $CellContext`zoom; Switch[$CellContext`solflag, 0, $CellContext`solPic = Graphics3D[{ Point[{0, 0, 0}]}]; Null, 1, $CellContext`solPic = \ $CellContext`plotPlane[{$CellContext`a1, $CellContext`b1, $CellContext`c1, \ $CellContext`d1}, $CellContext`range, Red]; Null, 2, $CellContext`solPic = \ $CellContext`plotIntersectionOf2Planes[{$CellContext`solx, $CellContext`soly, \ $CellContext`solz}, $CellContext`range, Red]; Null, 3, $CellContext`solPic = Graphics3D[{Red, Sphere[{$CellContext`solx, $CellContext`soly, \ $CellContext`solz}, $CellContext`range/16]}]; Null, 4, $CellContext`solPic = Graphics3D[{ Point[{0, 0, 0}]}]; Null]; $CellContext`axesPic = \ $CellContext`make3Daxes[$CellContext`range, White, White, White, "x", "y", "z", True]; $CellContext`surface1Pic = \ $CellContext`plotPlane[{$CellContext`a1, $CellContext`b1, $CellContext`c1, \ $CellContext`d1}, $CellContext`range, $CellContext`row1Color]; \ $CellContext`surface2Pic = $CellContext`plotPlane[{$CellContext`a2, \ $CellContext`b2, $CellContext`c2, $CellContext`d2}, $CellContext`range, \ $CellContext`row2Color]; $CellContext`surface3Pic = \ $CellContext`plotPlane[{$CellContext`a3, $CellContext`b3, $CellContext`c3, \ $CellContext`d3}, $CellContext`range, $CellContext`row3Color]; Show[$CellContext`axesPic, $CellContext`surface1Pic, \ $CellContext`surface2Pic, $CellContext`surface3Pic, $CellContext`solPic, Boxed -> False, ViewPoint -> {$CellContext`range Cos[$CellContext`view\[Theta]] Cos[$CellContext`view\[Phi]], $CellContext`range Sin[$CellContext`view\[Theta]] Cos[$CellContext`view\[Phi]], $CellContext`range Sin[$CellContext`view\[Phi]]}, ViewVertical -> {0, 0, 1}, ImageSize -> {500, 250}, SphericalRegion -> True, Lighting -> "Neutral", PlotRange -> {{(-1.1) $CellContext`range, 1.1 $CellContext`range}, {(-1.2) $CellContext`range, 1.2 $CellContext`range}, {(-1.2) $CellContext`range, 1.2 $CellContext`range}}]]}]]; Typeset`initDone$$ = True), SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellID->189201795], Cell[TextData[{ "The fundamental question in linear algebra is determining if solutions \ exist for a set of linear equations ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"A", ".", "x"}], "=", "b"}], TraditionalForm]], "InlineMath"], ". There are two ways to look at this problem. The first is the row view\ \[LongDash]treating the problem as finding the intersection of a set of \ planes defined by the equations. The second way is the column \ view\[LongDash]treating the problem as finding the solution of a linear \ combination of the column vectors of ", Cell[BoxData[ FormBox["A", TraditionalForm]], "InlineMath"], ". This Demonstration illustrates the problem from either viewpoint." }], "ManipulateCaption"], Cell["DETAILS", "DetailsSection"], Cell[TextData[{ "This Demonstration was created to accompany the first chapters in G. \ Strang, ", StyleBox["Introduction to Linear Algebra", FontSlant->"Italic"], ", 3rd ed., Wellesley, MA: Wellesley\[Dash]Cambridge Press, 2003." }], "DetailNotes", CellID->741297292], Cell["\<\ You can select examples of different types of systems of linear equations, or \ use the opener view and enter your own coefficients for the equations. \ \>", "DetailNotes", CellID->15113109], Cell["\<\ In the row interpretation, any solution appears in red. If there is a single \ solution, the point is shown by a red dot. If the planes intersect on a \ common line, the solution is shown as a red line. If the equations are \ coplanar, the solution is shown as a red plane.\ \>", "DetailNotes", CellID->15593668], Cell[TextData[{ "In each case, look at the column view to see what the corresponding column \ vectors of the matrix ", Cell[BoxData[ FormBox["A", TraditionalForm]], "InlineMath"], " look like. If these column vectors lie on the same plane, the plane is \ shown in brown, and there is a solution only if the vector ", Cell[BoxData[ FormBox["b", TraditionalForm]], "InlineMath"], " lies on the same plane. 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