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0.7000000000000002}, {2, 0.7000000000000002}}], Line[{{-2, 0.8000000000000003}, {2, 0.8000000000000003}}], Line[{{-2, 0.9000000000000004}, {2, 0.9000000000000004}}], Line[{{-2, 1.0000000000000004`}, {2, 1.0000000000000004`}}], Line[{{-2, 1.1}, {2, 1.1}}], Line[{{-2, 1.2000000000000002`}, {2, 1.2000000000000002`}}], Line[{{-2, 1.3000000000000003`}, {2, 1.3000000000000003`}}], Line[{{-2, 1.4000000000000004`}, {2, 1.4000000000000004`}}], Line[{{-2, 1.5000000000000004`}, {2, 1.5000000000000004`}}], Line[{{-2, 1.6}, {2, 1.6}}], Line[{{-2, 1.7000000000000002`}, {2, 1.7000000000000002`}}], Line[{{-2, 1.8000000000000003`}, {2, 1.8000000000000003`}}], Line[{{-2, 1.9000000000000004`}, {2, 1.9000000000000004`}}]}, GrayLevel[0.5], { Line[{{-1.5, -2}, {-1.5, 2}}], Line[{{-1., -2}, {-1., 2}}], Line[{{-0.5, -2}, {-0.5, 2}}], Line[{{0., -2}, {0., 2}}], Line[{{0.5, -2}, {0.5, 2}}], Line[{{1., -2}, {1., 2}}], Line[{{1.5, -2}, {1.5, 2}}]}, { Line[{{-2, -1.5}, {2, -1.5}}], Line[{{-2, -1.}, {2, -1.}}], Line[{{-2, -0.5}, {2, -0.5}}], Line[{{-2, 0.}, {2, 0.}}], Line[{{-2, 0.5}, {2, 0.5}}], Line[{{-2, 1.}, {2, 1.}}], Line[{{-2, 1.5}, {2, 1.5}}]}, GrayLevel[0], { Line[{{-1, -2}, {-1, 2}}], Line[{{0, -2}, {0, 2}}], Line[{{1, -2}, {1, 2}}]}, { Line[{{-2, -1}, {2, -1}}], Line[{{-2, 0}, {2, 0}}], Line[{{-2, 1}, {2, 1}}]}}}; ($CellContext`abc = {0, 0, -1}; $CellContext`def = {-0.3, 0.2, 0}; $CellContext`ghi = { 0, 0, 0.2}; $CellContext`lmn = {0.3, -0.45, 0}; $CellContext`opq[ Pattern[$CellContext`pt, Blank[]]] := (1/2) ( Cross[$CellContext`abc, $CellContext`pt - $CellContext`def]/ Dot[$CellContext`pt - $CellContext`def, $CellContext`pt - \ $CellContext`def] + Cross[$CellContext`ghi, $CellContext`pt - $CellContext`lmn]/ Dot[$CellContext`pt - $CellContext`lmn, $CellContext`pt - \ $CellContext`lmn]); $CellContext`bg = Polygon[{ ImageScaled[{0, 0}], ImageScaled[{1, 0}], ImageScaled[{1, 1}], ImageScaled[{0, 1}]}, VertexColors -> { LightGray, LightGreen, LightYellow, LightBlue}]; $CellContext`gridlines = {LightGray, Table[ Line[{{$CellContext`i, -2}, {$CellContext`i, 2}}], {$CellContext`i, -1.9, 1.9, 0.1}], Table[ Line[{{-2, $CellContext`i}, { 2, $CellContext`i}}], {$CellContext`i, -1.9, 1.9, 0.1}], Gray, Table[ Line[{{$CellContext`i, -2}, {$CellContext`i, 2}}], {$CellContext`i, -1.5, 1.5, 0.5}], Table[ Line[{{-2, $CellContext`i}, { 2, $CellContext`i}}], {$CellContext`i, -1.5, 1.5, 0.5}], Black, Table[ Line[{{$CellContext`i, -2}, {$CellContext`i, 2}}], {$CellContext`i, -1, 1, 1}], Table[ Line[{{-2, $CellContext`i}, { 2, $CellContext`i}}], {$CellContext`i, -1, 1, 1}]})}; Typeset`initDone$$ = True), SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellChangeTimes->{ 3.35696210375764*^9, 3.4149321637472143`*^9, 3.414932197215107*^9, { 3.414932235136011*^9, 3.414932252260573*^9}, 3.41493230061871*^9, 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Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> Graphics[{Black, Polygon[{{-0.5, 0.5}, {0.5, 0.5}, {0.5, -0.5}, {-0.5, -0.5}}], Red, If[ Or[Abs[ Part[$CellContext`p$$, 1]] > 0.5, Abs[ Part[$CellContext`p$$, 2]] > 0.5], Arrow[{$CellContext`p$$, $CellContext`p$$ + { Part[ $CellContext`opq[{ Part[$CellContext`p$$, 1], Part[$CellContext`p$$, 2], 0}], 1], Part[ $CellContext`lklj[{ Part[$CellContext`p$$, 1], Part[$CellContext`p$$, 2], 0}], 2]}}], PointSize[0.02]], Gray, Circle[$CellContext`p$$, 1], Circle[$CellContext`p$$, 2], Circle[$CellContext`p$$, 0.5], Circle[$CellContext`p$$, 1.5], Lighter[ Lighter[Gray]], Circle[$CellContext`p$$, 0.1], Circle[$CellContext`p$$, 0.2], Circle[$CellContext`p$$, 0.3], Circle[$CellContext`p$$, 0.4], Circle[$CellContext`p$$, 0.6], Circle[$CellContext`p$$, 0.7], Circle[$CellContext`p$$, 0.8], Circle[$CellContext`p$$, 0.9], Circle[$CellContext`p$$, 1.1], Circle[$CellContext`p$$, 1.2], Circle[$CellContext`p$$, 1.3], Circle[$CellContext`p$$, 1.4], Circle[$CellContext`p$$, 1.6], Circle[$CellContext`p$$, 1.7], Circle[$CellContext`p$$, 1.8], Circle[$CellContext`p$$, 1.9], Gray, Table[ Line[{$CellContext`p$$ + 0.95 { Cos[$CellContext`i], Sin[$CellContext`i]}, $CellContext`p$$ + 1.05 { Cos[$CellContext`i], Sin[$CellContext`i]}}], {$CellContext`i, 0, 2 Pi, 2 (Pi/36)}]}, GridLines -> {{-1, 0, 1}, {-1, 0, 1}}, Frame -> True, PlotRange -> {{-2, 2}, {-2, 2}}, Prolog -> {$CellContext`bg, $CellContext`gridlines}], "Specifications" :> {{{$CellContext`p$$, {0.32, -0.51}}, Automatic, ControlType -> Locator}}, "Options" :> {}, "DefaultOptions" :> {ControllerLinking -> True}], ImageSizeCache->{406., {207., 212.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, Initialization:>({{$CellContext`opq[ Pattern[$CellContext`pt, Blank[]]] := (1/2) ( Cross[$CellContext`abc, $CellContext`pt - $CellContext`def]/ Dot[$CellContext`pt - $CellContext`def, $CellContext`pt - \ $CellContext`def] + Cross[$CellContext`ghi, $CellContext`pt - $CellContext`lmn]/ Dot[$CellContext`pt - $CellContext`lmn, $CellContext`pt - \ $CellContext`lmn]), $CellContext`abc = {0, 0, -1}, $CellContext`def = {-0.3, 0.2, 0}, $CellContext`ghi = { 0, 0, 0.2}, $CellContext`lmn = {0.3, -0.45, 0}, $CellContext`lklj[ Pattern[$CellContext`t, Blank[]]] := {(1/ 2) ((0.09000000000000001 - 0.2 Part[$CellContext`t, 2])/((-0.3 + Part[$CellContext`t, 1])^2 + (-0.45 + Part[$CellContext`t, 2])^2 + Part[$CellContext`t, 3]^2) + (0.2 + Part[$CellContext`t, 2])/((0.3 + Part[$CellContext`t, 1])^2 + ( 0.2 + Part[$CellContext`t, 2])^2 + Part[$CellContext`t, 3]^2)), (1/ 2) ((-0.06 + 0.2 Part[$CellContext`t, 1])/((-0.3 + Part[$CellContext`t, 1])^2 + (-0.45 + Part[$CellContext`t, 2])^2 + Part[$CellContext`t, 3]^2) + (-0.3 - Part[$CellContext`t, 1])/((0.3 + Part[$CellContext`t, 1])^2 + ( 0.2 + Part[$CellContext`t, 2])^2 + Part[$CellContext`t, 3]^2)), 0}, $CellContext`bg = Polygon[{ ImageScaled[{0, 0}], ImageScaled[{1, 0}], ImageScaled[{1, 1}], ImageScaled[{0, 1}]}, VertexColors -> { GrayLevel[0.85], RGBColor[0.88, 1, 0.88], RGBColor[1, 1, 0.85], RGBColor[0.87, 0.94, 1]}], $CellContext`gridlines = { GrayLevel[0.85], { Line[{{-1.9, -2}, {-1.9, 2}}], Line[{{-1.7999999999999998`, -2}, {-1.7999999999999998`, 2}}], Line[{{-1.7, -2}, {-1.7, 2}}], Line[{{-1.5999999999999999`, -2}, {-1.5999999999999999`, 2}}], Line[{{-1.5, -2}, {-1.5, 2}}], Line[{{-1.4, -2}, {-1.4, 2}}], Line[{{-1.2999999999999998`, -2}, {-1.2999999999999998`, 2}}], Line[{{-1.1999999999999997`, -2}, {-1.1999999999999997`, 2}}], Line[{{-1.0999999999999999`, -2}, {-1.0999999999999999`, 2}}], Line[{{-0.9999999999999999, -2}, {-0.9999999999999999, 2}}], Line[{{-0.8999999999999999, -2}, {-0.8999999999999999, 2}}], Line[{{-0.7999999999999998, -2}, {-0.7999999999999998, 2}}], Line[{{-0.6999999999999997, -2}, 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Line[{{1.5, -2}, {1.5, 2}}]}, { Line[{{-2, -1.5}, {2, -1.5}}], Line[{{-2, -1.}, {2, -1.}}], Line[{{-2, -0.5}, {2, -0.5}}], Line[{{-2, 0.}, {2, 0.}}], Line[{{-2, 0.5}, {2, 0.5}}], Line[{{-2, 1.}, {2, 1.}}], Line[{{-2, 1.5}, {2, 1.5}}]}, GrayLevel[0], { Line[{{-1, -2}, {-1, 2}}], Line[{{0, -2}, {0, 2}}], Line[{{1, -2}, {1, 2}}]}, { Line[{{-2, -1}, {2, -1}}], Line[{{-2, 0}, {2, 0}}], Line[{{-2, 1}, {2, 1}}]}}}; ($CellContext`abc = {0, 0, -1}; $CellContext`def = {-0.3, 0.2, 0}; $CellContext`ghi = { 0, 0, 0.2}; $CellContext`lmn = {0.3, -0.45, 0}; $CellContext`opq[ Pattern[$CellContext`pt, Blank[]]] := (1/2) ( Cross[$CellContext`abc, $CellContext`pt - $CellContext`def]/ Dot[$CellContext`pt - $CellContext`def, $CellContext`pt - \ $CellContext`def] + Cross[$CellContext`ghi, $CellContext`pt - $CellContext`lmn]/ Dot[$CellContext`pt - $CellContext`lmn, $CellContext`pt - \ $CellContext`lmn]); $CellContext`bg = Polygon[{ ImageScaled[{0, 0}], ImageScaled[{1, 0}], ImageScaled[{1, 1}], ImageScaled[{0, 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solving a practical real-world mystery." }], "DetailNotes", CellChangeTimes->{ 3.35696210375764*^9, {3.4150148511624765`*^9, 3.415014902348666*^9}, { 3.415014940503939*^9, 3.4150149462381673`*^9}, {3.4150150054554014`*^9, 3.415015088562649*^9}, {3.415015137623893*^9, 3.4150152292777967`*^9}, { 3.4150152686517887`*^9, 3.415015354962079*^9}, {3.4150239081962357`*^9, 3.415023940523533*^9}}, CellID->206439416] }, Open ]], Cell[CellGroupData[{ Cell["", "ControlSuggestionsSection"], Cell[BoxData[ TooltipBox[ RowBox[{ CheckboxBox[False], Cell[" Resize Images"]}], "\"Click inside an image to reveal its orange resize frame.\\nDrag any of \ the orange resize handles to resize the image.\"", ActionDelay->0.35]], "ControlSuggestions", FontFamily->"Verdana", CellTags->"ResizeImages"], Cell[BoxData[ TooltipBox[ RowBox[{ CheckboxBox[False], Cell[" Rotate and Zoom in 3D"]}], RowBox[{ "\"Drag a 3D graphic to rotate it. Starting the drag near the center \ tumbles\\nthe graphic; starting near a corner turns it parallel to the plane \ of the screen.\\nHold down \"", FrameBox[ "Ctrl", Background -> GrayLevel[0.9], FrameMargins -> 2, FrameStyle -> GrayLevel[0.9]], "\" (or \"", FrameBox[ "Cmd", Background -> GrayLevel[0.9], FrameMargins -> 2, FrameStyle -> GrayLevel[0.9]], "\" on Mac) and drag up and down to zoom.\""}], ActionDelay->0.35]], "ControlSuggestions", FontFamily->"Verdana", CellTags->"RotateAndZoomIn3D"], Cell[BoxData[ TooltipBox[ RowBox[{ CheckboxBox[False], Cell[" Drag Locators"]}], RowBox[{"\"Drag any locator (\"", GraphicsBox[ LocatorBox[ Scaled[{0.5, 0.5}]], ImageSize -> 20], "\", etc.) to move it around.\""}], ActionDelay->0.35]], "ControlSuggestions", FontFamily->"Verdana", CellTags->"DragLocators"], Cell[BoxData[ TooltipBox[ RowBox[{ CheckboxBox[False], Cell[" Create and Delete Locators"]}], RowBox[{"\"Insert a new locator in the graphic by holding down the \"", FrameBox[ "Alt", Background -> GrayLevel[0.9], FrameMargins -> 2, FrameStyle -> GrayLevel[0.9]], "\" key\\nand clicking where you want it to be. 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