(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 66885, 2245] NotebookOptionsPosition[ 60866, 2065] NotebookOutlinePosition[ 61958, 2105] CellTagsIndexPosition[ 61858, 2099] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell["Wright State Calculus Laboratory Project", "Subsubtitle"], Cell[CellGroupData[{ Cell["56 Rotations", "Title"], Cell[CellGroupData[{ Cell["About This Notebook", "Subsection", CellChangeTimes->{{3.436697925660022*^9, 3.436697927018022*^9}}], Cell[BoxData[ RowBox[{"Needs", "[", "\"\\"", "]"}]], "Input", InitializationCell->True, CellChangeTimes->{{3.407853210189947*^9, 3.4078532206741743`*^9}, { 3.407853282377083*^9, 3.407853287023775*^9}, {3.410883377377009*^9, 3.410883379578103*^9}, 3.43669461170588*^9}], Cell["\<\ This notebook requires the package calcE, version 7.0 or higher.\ \>", "Text", CellChangeTimes->{3.397935087111536*^9, 3.4366944917170877`*^9}], Cell["\<\ Version 10.1, 9/1/2009. Written by Richard Mercer, Copyright 2009.\ \>", "Text", CellChangeTimes->{{3.3876375566401863`*^9, 3.38763757395231*^9}, { 3.39608587621196*^9, 3.396085876409902*^9}, {3.397935124815791*^9, 3.3979351324220943`*^9}, 3.407356633446836*^9, 3.4157873998634863`*^9, 3.4184006862452087`*^9, {3.423862718096117*^9, 3.423862718270091*^9}, { 3.4366949572682*^9, 3.4366949615548277`*^9}, {3.459171844669262*^9, 3.459171855770858*^9}}] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Rotations in the Plane", "Subtitle"], Cell[CellGroupData[{ Cell["Rotating Vectors", "Section"], Cell[TextData[{ "\tSuppose we have the vector ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox[ SubscriptBox[ StyleBox["v", FontWeight->"Bold", FontSlant->"Plain"], StyleBox["1", FontWeight->"Plain"]], FontWeight->"Bold", FontSlant->"Plain"], "=", RowBox[{"(", RowBox[{"1", ",", "3"}], ")"}]}], TraditionalForm]]], " in the plane and want to rotate it about the origin through an angle of ", Cell[BoxData[ FormBox[ FractionBox["\[Pi]", "4"], TraditionalForm]]], " radians (counterclockwise). Where will it end up?" }], "Text", CellChangeTimes->{{3.421092977109375*^9, 3.42109302946875*^9}, { 3.43031575249649*^9, 3.4303157631907063`*^9}, 3.430315797512267*^9}], Cell[BoxData[{ RowBox[{ RowBox[{"v1", "=", RowBox[{"{", RowBox[{"1", ",", "3"}], "}"}]}], ";"}], "\n", RowBox[{"PlotE", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "4"}], ",", "4"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "4"}], "}"}], ",", "Same", ",", RowBox[{"Point", "[", RowBox[{"v1", ",", "Red", ",", "Coords", ",", "Above"}], "]"}], ",", RowBox[{"Point", "[", RowBox[{ RowBox[{"RotatE", "[", RowBox[{"v1", ",", FractionBox["\[Pi]", "4"]}], "]"}], ",", "Red", ",", "\"\<(??,??)\>\"", ",", "Above"}], "]"}], ",", RowBox[{"Vector", "[", RowBox[{"v1", ",", "Red"}], "]"}], ",", RowBox[{"Vector", "[", RowBox[{ RowBox[{"RotatE", "[", RowBox[{"v1", ",", FractionBox["\[Pi]", "4"]}], "]"}], ",", "Red"}], "]"}]}], "]"}]}], "Input", CellChangeTimes->{ 3.391888868104093*^9, {3.430306362292124*^9, 3.430306411950798*^9}, { 3.4313394967328167`*^9, 3.43133949800966*^9}, {3.436696040000812*^9, 3.436696042535206*^9}}], Cell["Polar Coordinates to the rescue!", "Subsection"], Cell["\<\ \tPolar coordinates are ideal for dealing with rotations. Our strategy:\ \>", "Text", CellChangeTimes->{{3.421093037859375*^9, 3.421093063375*^9}, { 3.430315800556066*^9, 3.4303158039560747`*^9}, {3.436696079135188*^9, 3.436696098645294*^9}, {3.4371507389723663`*^9, 3.437150742433834*^9}, 3.437151015518614*^9}], Cell["convert the point to polar coordinates.", "Item", CellChangeTimes->{{3.421093037859375*^9, 3.421093063375*^9}, { 3.430315800556066*^9, 3.4303158039560747`*^9}, {3.436696079135188*^9, 3.436696098645294*^9}, {3.4371507389723663`*^9, 3.437150742433834*^9}, { 3.437151015518614*^9, 3.437151040914742*^9}}], Cell[TextData[{ "add ", Cell[BoxData[ FormBox[ FractionBox["\[Pi]", "4"], TraditionalForm]]], " to the angle variable ", Cell[BoxData[ FormBox["\[Theta]", TraditionalForm]]], "." }], "Item", CellChangeTimes->{{3.421093037859375*^9, 3.421093063375*^9}, { 3.430315800556066*^9, 3.4303158039560747`*^9}, {3.436696079135188*^9, 3.436696098645294*^9}, {3.4371507389723663`*^9, 3.437150742433834*^9}, { 3.437151015518614*^9, 3.437151043338793*^9}}], Cell["convert back to rectangular coordinates.", "Item", CellChangeTimes->{{3.421093037859375*^9, 3.421093063375*^9}, { 3.430315800556066*^9, 3.4303158039560747`*^9}, {3.436696079135188*^9, 3.436696098645294*^9}, {3.4371507389723663`*^9, 3.437150742433834*^9}, { 3.437151015518614*^9, 3.43715104647472*^9}}], Cell[TextData[{ "\t", StyleBox["Important", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], ": This procedure is the conceptual basis for the rotation formula \ introduced in the next section. The best way to rotate is to use ", ButtonBox["that formula", BaseStyle->"Hyperlink", ButtonData->"Formula"], ", ", StyleBox["not", FontWeight->"Bold", FontSlant->"Italic"], " to imitate what we do here!" }], "Text", CellChangeTimes->{{3.4371507476096907`*^9, 3.437150910699019*^9}, { 3.437150987077427*^9, 3.437150987077874*^9}}], Cell[TextData[{ "\tWe begin by converting the original point to polar coordinates ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ SubscriptBox["r", "1"], ",", SubscriptBox["\[Theta]", "1"]}], ")"}], TraditionalForm]]], "." }], "Text", CellChangeTimes->{{3.421093037859375*^9, 3.421093063375*^9}, { 3.430315800556066*^9, 3.4303158039560747`*^9}, {3.436696079135188*^9, 3.436696094733955*^9}}], Cell[BoxData[ RowBox[{"RectToPolar", "[", "v1", "]"}]], "Input", CellChangeTimes->{{3.430317687535677*^9, 3.430317704340002*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"{", RowBox[{"a1", ",", "b1"}], "}"}], "=", RowBox[{"{", RowBox[{"1", ",", "3"}], "}"}]}], ";"}], "\n", RowBox[{ RowBox[{"{", RowBox[{"r1", ",", "\[Theta]1"}], "}"}], "=", RowBox[{"{", RowBox[{ RowBox[{"Norm", "[", RowBox[{"{", RowBox[{"a1", ",", "b1"}], "}"}], "]"}], ",", RowBox[{"ArcTan", "[", RowBox[{"a1", ",", "b1"}], "]"}]}], "}"}]}]}], "Input", CellChangeTimes->{{3.4303175881329527`*^9, 3.430317610556402*^9}, { 3.4303176560350943`*^9, 3.4303176606417503`*^9}}], Cell[TextData[{ "\tTo rotate by ", Cell[BoxData[ FormBox[ FractionBox["\[Pi]", "4"], TraditionalForm]]], ", all you have to do is add ", Cell[BoxData[ FormBox[ FractionBox["\[Pi]", "4"], TraditionalForm]]], " onto ", Cell[BoxData[ FormBox[ SubscriptBox["\[Theta]", "1"], TraditionalForm]]], " and leave the length unchanged. The second point will have polar \ coordinates ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"(", RowBox[{ SubscriptBox["r", "2"], ",", SubscriptBox["\[Theta]", "2"]}], ")"}], "=", RowBox[{"(", RowBox[{ SubscriptBox["r", "1"], ",", RowBox[{ SubscriptBox["\[Theta]", "1"], "+", FractionBox["\[Pi]", "4"]}]}], ")"}]}], TraditionalForm]]], ". " }], "Text", CellChangeTimes->{{3.421093075625*^9, 3.42109317325*^9}, 3.430315810792014*^9}], Cell[BoxData[ RowBox[{ RowBox[{"{", RowBox[{"r2", ",", "\[Theta]2"}], "}"}], "=", RowBox[{"{", RowBox[{"r1", ",", RowBox[{"\[Theta]1", "+", FractionBox["\[Pi]", "4"]}]}], "}"}]}]], "Input"], Cell[TextData[{ "\tNow convert back to rectangular coordinates ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ SubscriptBox["a", "2"], ",", SubscriptBox["b", "2"]}], ")"}], TraditionalForm]]], ", and call this the vector ", Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox[ StyleBox["v", FontWeight->"Bold", FontSlant->"Plain"], StyleBox["2", FontWeight->"Plain"]], FontWeight->"Bold", FontSlant->"Plain"], TraditionalForm]]], ". " }], "Text", CellChangeTimes->{{3.421093219765625*^9, 3.42109324034375*^9}, { 3.430315814116858*^9, 3.430315824605204*^9}, {3.430317512174138*^9, 3.4303175165275373`*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{"v2", " ", "=", RowBox[{"{", RowBox[{ RowBox[{"r2", " ", RowBox[{"Cos", "[", "\[Theta]2", "]"}]}], ",", RowBox[{"r2", " ", RowBox[{"Sin", "[", "\[Theta]2", "]"}]}]}], "}"}]}], " "}], "\[IndentingNewLine]", RowBox[{"v2", " ", "=", RowBox[{"v2", " ", "//", "Simplify"}]}]}], "Input", CellChangeTimes->{{3.437150529680492*^9, 3.437150531870356*^9}, { 3.437150564820408*^9, 3.437150571514681*^9}, {3.437150650660774*^9, 3.437150682011011*^9}}], Cell[TextData[{ "\tYou have rotated the vector counterclockwise by ", Cell[BoxData[ FormBox[ FractionBox["\[Pi]", "4"], TraditionalForm]]], " radians." }], "Text", CellChangeTimes->{{3.421093274234375*^9, 3.421093277296875*^9}, 3.430315826844995*^9}], Cell[BoxData[ RowBox[{"PlotE", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "4"}], ",", "4"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "4"}], "}"}], ",", "Same", ",", RowBox[{"{", RowBox[{"Red", ",", RowBox[{"Point", "[", RowBox[{"v1", ",", "v2", ",", "Coords", ",", "Above"}], "]"}], ",", RowBox[{"Vector", "[", RowBox[{"v1", ",", "\"\\"", ",", "Right"}], "]"}], ",", RowBox[{"Vector", "[", RowBox[{"v2", ",", "\"\\"", ",", "Left"}], "]"}]}], "}"}]}], "]"}]], "Input", CellChangeTimes->{3.391888869327141*^9, 3.431339530403399*^9}] }, Open ]], Cell[CellGroupData[{ Cell["The Rotation Formula", "Section"], Cell[TextData[{ "\tRather than going through the above three-step procedure every time you \ want to rotate something, let\[CloseCurlyQuote]s use that procedure to find a \ formula.\n\tUsing polar coordinates, you can find formulas for rotating \ points. Suppose your original point has rectangular coordinates ", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{"(", RowBox[{ SubscriptBox["x", "1"], ",", SubscriptBox["y", "1"]}], ")"}], FontColor->RGBColor[1, 0, 0]], TraditionalForm]]], " and polar coordinates ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ SubscriptBox["r", "1"], ",", SubscriptBox["\[Theta]", "1"]}], ")"}], TraditionalForm]], FontColor->RGBColor[0, 0, 1]], ", and you want to rotate this point by an angle ", StyleBox["\[Theta]", FontColor->RGBColor[0, 0, 1]], "." }], "Text", CellChangeTimes->{{3.4210932895*^9, 3.421093319015625*^9}, { 3.4275773420685577`*^9, 3.427577381959681*^9}, {3.430315829894989*^9, 3.4303158316936607`*^9}, {3.4369886099995537`*^9, 3.4369886099996157`*^9}}], Cell[BoxData[ GraphicsBox[{}, AspectRatio->Automatic, Axes->True, AxesOrigin->Automatic, Background->RGBColor[1, 1, 0.75], Epilog->{{ RGBColor[0, 0, 1], {{ LineBox[{{0, 0}, {0.3623577544766736, 0.9320390859672263}}], InsetBox[ FormBox[ StyleBox[ "\"\\!\\(\\*SubscriptBox[\\(r\\), \\(1\\)]\\)\"", FontSize -> 14, FontSlant -> Italic, StripOnInput -> False], TraditionalForm], Scaled[{0.03, 0.}, {0.1811788772383368, 0.46601954298361314`}], {-1, 0}]}, { LineBox[{{0, 0}, {-0.5885011172553458, 0.8084964038195901}}], InsetBox[ FormBox[ StyleBox[ "\"\\!\\(\\*SubscriptBox[\\(r\\), \\(1\\)]\\)\"", FontSize -> 14, FontSlant -> Italic, StripOnInput -> False], TraditionalForm], Scaled[{0.03, 0.}, {-0.2942505586276729, 0.40424820190979505`}], {-1, 0}]}, { CircleBox[{0., 0.}, 0.3, {0., 2.2}]}, { InsetBox[ FormBox[ StyleBox[ "\"\\!\\(TraditionalForm\\`\\\"(\\\\!\\\\(\\\\*SubscriptBox[\\\\(r\\\ \\), \\\\(1\\\\)]\\\\), \\\\!\\\\(\\\\*SubscriptBox[\\\\(\\\\[Theta]\\\\), \\\ \\(1\\\\)]\\\\))\\\"\\)\"", FontSize -> 14, StripOnInput -> False], TraditionalForm], Scaled[{-0.03, 0.}, {0.3623577544766736, 0.9320390859672263}], {1, 0}]}, { InsetBox[ FormBox[ StyleBox[ "\"\\!\\(TraditionalForm\\`\\\"(\\\\!\\\\(\\\\*SubscriptBox[\\\\(r\\\ \\), \\\\(2\\\\)]\\\\), \\\\!\\\\(\\\\*SubscriptBox[\\\\(\\\\[Theta]\\\\), \\\ \\(2\\\\)]\\\\))\\\"\\)\"", FontSize -> 14, StripOnInput -> False], TraditionalForm], Scaled[{-0.03, 0.}, {-0.5885011172553458, 0.8084964038195901}], {1, 0}]}, { InsetBox[ FormBox[ StyleBox[ "\"\\!\\(TraditionalForm\\`\\\"\\\\!\\\\(\\\\*SubscriptBox[\\\\(\\\\[\ Theta]\\\\), \\\\(1\\\\)]\\\\)\\\"\\)\"", FontSize -> 14, StripOnInput -> False], TraditionalForm], {0.3, 0.2}]}, { InsetBox[ FormBox[ StyleBox[ "\"\\!\\(TraditionalForm\\`\\\"\\\\[Theta]\\\"\\)\"", FontSize -> 14, StripOnInput -> False], TraditionalForm], {-0.05, 0.35}]}}}, { RGBColor[1, 0, 0], { InsetBox[ FormBox[ StyleBox[ "\"(\\!\\(x\\_1\\), \\!\\(y\\_1\\))\"", FontSize -> 14, StripOnInput -> False], TraditionalForm], Scaled[{0.03, 0.}, {0.3623577544766736, 0.9320390859672263}], {-1, 0}], PointSize[Medium], TagBox[ TooltipBox[ PointBox[{0.3623577544766736, 0.9320390859672263}], "\"(0.362358, 0.932039)\"", LabelStyle -> {}], Annotation[#, "(0.362358, 0.932039)", "Tooltip"]& ]}, { InsetBox[ FormBox[ StyleBox[ "\"(\\!\\(x\\_2\\), \\!\\(y\\_2\\))\"", FontSize -> 14, StripOnInput -> False], TraditionalForm], Scaled[{0.03, 0.}, {-0.5885011172553458, 0.8084964038195901}], {-1, 0}], PointSize[Medium], TagBox[ TooltipBox[ PointBox[{-0.5885011172553458, 0.8084964038195901}], "\"(-0.588501, 0.808496)\"", LabelStyle -> {}], Annotation[#, "(-0.588501, 0.808496)", "Tooltip"]& ]}}}, Frame->False, PlotRange->{{-1, 1}, {0, 1}}, PlotRangePadding->Scaled[0.02], Prolog->{}, Ticks->None]], "Output", CellChangeTimes->{ 3.4369893811714087`*^9, 3.436989670088442*^9, 3.436989705793165*^9, { 3.436989982420766*^9, 3.436990039606141*^9}, 3.436991606845172*^9, { 3.436991644216978*^9, 3.436991718160165*^9}, {3.436991808918261*^9, 3.436991832525832*^9}, {3.436991940121482*^9, 3.436992065083838*^9}, 3.4369921174214497`*^9, {3.4369923054349403`*^9, 3.436992320411859*^9}}, TextAlignment->Center], Cell[TextData[{ "\tThe radius is not changed by the rotation, but the angle is increased by \ \[Theta]. The new polar coordinates will be ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"(", RowBox[{ SubscriptBox["r", "2"], ",", SubscriptBox["\[Theta]", "2"]}], ")"}], "=", RowBox[{"(", RowBox[{ SubscriptBox["r", "1"], ",", RowBox[{ SubscriptBox["\[Theta]", "1"], "+", "\[Theta]"}]}], ")"}]}], TraditionalForm]]], ", so the new rectangular coordinates will be ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"(", RowBox[{ SubscriptBox["x", "2"], ",", SubscriptBox["y", "2"]}], ")"}], "="}], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ RowBox[{ SubscriptBox["r", "1"], " ", RowBox[{"cos", "(", RowBox[{ SubscriptBox["\[Theta]", "1"], "+", "\[Theta]"}], ")"}]}], ",", RowBox[{ SubscriptBox["r", "1"], " ", RowBox[{"sin", "(", RowBox[{ SubscriptBox["\[Theta]", "1"], "+", "\[Theta]"}], ")"}]}]}], ")"}], TraditionalForm]]], ". \n\tWe can then use the addition identities for sine and cosine:" }], "Text", CellChangeTimes->{{3.42109333525*^9, 3.421093386625*^9}, { 3.4275774494564857`*^9, 3.4275774693574142`*^9}, 3.4303158382138367`*^9, { 3.4369886306817102`*^9, 3.436988691070397*^9}, {3.4369921512838097`*^9, 3.436992159461564*^9}, {3.436992335762136*^9, 3.436992360314002*^9}, { 3.436992462893777*^9, 3.436992525196632*^9}}], Cell[TextData[{ "You ", StyleBox["do ", FontWeight->"Bold"], "remember them, don\[CloseCurlyQuote]t you? This is their big moment!" }], "SmallText"], Cell[TextData[{ Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["x", "2"], "=", RowBox[{ RowBox[{ SubscriptBox["r", "1"], " ", RowBox[{"cos", "(", RowBox[{ SubscriptBox["\[Theta]", "1"], "+", "\[Theta]"}], ")"}]}], "=", RowBox[{ RowBox[{ SubscriptBox["r", "1"], "(", " ", RowBox[{ RowBox[{ RowBox[{"cos", "(", SubscriptBox["\[Theta]", "1"], ")"}], RowBox[{"cos", "(", "\[Theta]", ")"}]}], "-", RowBox[{ RowBox[{"sin", "(", SubscriptBox["\[Theta]", "1"], ")"}], RowBox[{"sin", "(", "\[Theta]", ")"}]}]}], ")"}], "=", RowBox[{ RowBox[{ StyleBox[ SubscriptBox["r", "1"], FontColor->RGBColor[1, 0, 0]], StyleBox[" ", FontColor->RGBColor[1, 0, 0]], StyleBox[ RowBox[{"cos", "(", SubscriptBox["\[Theta]", "1"], ")"}], FontColor->RGBColor[1, 0, 0]], RowBox[{"cos", "(", "\[Theta]", ")"}]}], "-", RowBox[{ StyleBox[ SubscriptBox["r", "1"], FontColor->RGBColor[1, 0, 0]], StyleBox[" ", FontColor->RGBColor[1, 0, 0]], StyleBox[ RowBox[{"sin", "(", SubscriptBox["\[Theta]", "1"], ")"}], FontColor->RGBColor[1, 0, 0]], RowBox[{"sin", "(", "\[Theta]", ")"}]}]}]}]}]}], TraditionalForm]]], "\n", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["y", "2"], "=", RowBox[{ RowBox[{ SubscriptBox["r", "1"], " ", "sin", " ", RowBox[{"(", RowBox[{ SubscriptBox["\[Theta]", "1"], "+", "\[Theta]"}], ")"}]}], "=", RowBox[{ RowBox[{ SubscriptBox["r", "1"], "(", RowBox[{ RowBox[{ RowBox[{"sin", "(", SubscriptBox["\[Theta]", "1"], ")"}], RowBox[{"cos", "(", "\[Theta]", ")"}]}], "+", " ", RowBox[{ RowBox[{"cos", "(", SubscriptBox["\[Theta]", "1"], ")"}], RowBox[{"sin", "(", "\[Theta]", ")"}]}]}], ")"}], "=", RowBox[{ RowBox[{ StyleBox[ SubscriptBox["r", "1"], FontColor->RGBColor[1, 0, 0]], StyleBox[" ", FontColor->RGBColor[1, 0, 0]], StyleBox[ RowBox[{"sin", "(", SubscriptBox["\[Theta]", "1"], ")"}], FontColor->RGBColor[1, 0, 0]], RowBox[{"cos", "(", "\[Theta]", ")"}]}], "+", RowBox[{ StyleBox[ SubscriptBox["r", "1"], FontColor->RGBColor[1, 0, 0]], StyleBox[ RowBox[{"cos", "(", SubscriptBox["\[Theta]", "1"], ")"}], FontColor->RGBColor[1, 0, 0]], RowBox[{"sin", "(", "\[Theta]", ")"}]}]}]}]}]}], TraditionalForm]]] }], "Text", CellChangeTimes->{{3.421093393890625*^9, 3.421093552890625*^9}, { 3.4275773975926723`*^9, 3.427577404811434*^9}, {3.4275774856013002`*^9, 3.427577510848487*^9}, {3.436992213084893*^9, 3.436992238300111*^9}, { 3.436992412500408*^9, 3.436992450562705*^9}, {3.4369926077809143`*^9, 3.4369926206043997`*^9}, {3.4369926519198723`*^9, 3.436992723678335*^9}, { 3.436992762751568*^9, 3.436992766751947*^9}, {3.436992856995596*^9, 3.4369928569956417`*^9}}], Cell[TextData[{ "\tWe can now use our polar coordinate formulas ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["x", "1"], "=", RowBox[{"r", " ", RowBox[{"cos", "(", SubscriptBox["\[Theta]", "1"], ")"}]}]}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["y", "1"], "=", RowBox[{"r", " ", RowBox[{"sin", "(", SubscriptBox["\[Theta]", "1"], ")"}]}]}], TraditionalForm]]], " to get:" }], "Text", CellChangeTimes->{{3.42109356965625*^9, 3.421093589875*^9}, { 3.427577514187345*^9, 3.42757751676061*^9}, 3.430315841237855*^9, { 3.436992790840942*^9, 3.436992808781836*^9}}], Cell[TextData[{ "Rotation Formulas\n", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["x", "2"], "=", RowBox[{ RowBox[{ SubscriptBox["x", "1"], RowBox[{"cos", "(", "\[Theta]", ")"}]}], "-", RowBox[{ SubscriptBox["y", "1"], RowBox[{"sin", "(", "\[Theta]", ")"}]}]}]}], TraditionalForm]]], "\n", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["y", "2"], "=", RowBox[{ RowBox[{ SubscriptBox["x", "1"], RowBox[{"sin", "(", "\[Theta]", ")"}]}], "+", RowBox[{ SubscriptBox["y", "1"], RowBox[{"cos", "(", "\[Theta]", ")"}]}]}]}], TraditionalForm]]] }], "DisplayText", CellChangeTimes->{{3.421093593640625*^9, 3.421093631109375*^9}}, FontWeight->"Bold", CellTags->"Formula"] }, Open ]], Cell["Backwards", "Subsection", CellTags->"Backwards"], Cell["\<\ \tWhat if you want to go backwards, that is clockwise instead of \ counterclockwise? It\[CloseCurlyQuote]s easy; just use a negative angle in \ the formula!\ \>", "Text", CellChangeTimes->{{3.421093644953125*^9, 3.421093727453125*^9}, { 3.430315846718443*^9, 3.430315850829808*^9}, {3.436992946507805*^9, 3.436993009283984*^9}}], Cell[CellGroupData[{ Cell["Rotating Graphs", "Section"], Cell["Parametric Equations", "Subsection"], Cell["\<\ \tLet\[CloseCurlyQuote]s graph a typical ellipse using parametric equations.\ \>", "Text", CellChangeTimes->{3.430315867542367*^9}], Cell[TextData[Cell[BoxData[ FormBox[ RowBox[{ RowBox[{GridBox[{ { RowBox[{"x", "=", RowBox[{"2", RowBox[{"cos", "(", "t", ")"}]}]}]}, { RowBox[{"y", "=", RowBox[{"3", RowBox[{"sin", "(", "t", ")"}]}]}]} }], "0"}], "\[LessEqual]", "t", "\[LessEqual]", RowBox[{"2", "\[Pi]"}]}], TraditionalForm]]]], "DisplayFormula", CellChangeTimes->{{3.4210939183125*^9, 3.421093962703125*^9}}, GridBoxOptions->{ GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}}], Cell[BoxData[ RowBox[{"ellipse1", "=", RowBox[{"PlotE", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"2", " ", RowBox[{"Cos", "[", "t", "]"}]}], ",", RowBox[{"3", " ", RowBox[{"Sin", "[", "t", "]"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", RowBox[{"2", " ", "\[Pi]"}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "4"}], ",", "4"}], "}"}], ",", "Square"}], "]"}]}]], "Input", CellChangeTimes->{3.391888870443418*^9}], Cell[TextData[{ "\tIt\[CloseCurlyQuote]s a good idea to use symmetric intervals and ", StyleBox["Square", "Code"], " or ", StyleBox["Same", "Code"], " when doing rotations. Rotations don\[CloseCurlyQuote]t look right if \ different scales are used on the axes." }], "Text", CellChangeTimes->{{3.431339609105483*^9, 3.4313396649121113`*^9}}], Cell[TextData[{ "\tNow let\[CloseCurlyQuote]s rotate this ellipse ", Cell[BoxData[ FormBox[ FractionBox["\[Pi]", "6"], TraditionalForm]]], " radians counterclockwise. " }], "Text", CellChangeTimes->{{3.42109397584375*^9, 3.4210939790625*^9}, 3.430316017300941*^9, 3.431339687573284*^9}], Cell[TextData[{ "\tWhat are the parametric equations of the new ellipse? We can find them \ applying our rotation equations to the original parametric equations ", Cell[BoxData[ FormBox[GridBox[{ { RowBox[{ SubscriptBox["x", "1"], "=", RowBox[{"2", RowBox[{"cos", "(", "t", ")"}]}]}]}, { RowBox[{ SubscriptBox["y", "1"], "=", RowBox[{"3", RowBox[{"sin", "(", "t", ")"}]}]}]} }], TraditionalForm]]], ". 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What equation do the new points satisfy? We know that the old point ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ SubscriptBox["x", "1"], ",", SubscriptBox["y", "1"]}], ")"}], TraditionalForm]]], " satisfies the equation ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"9", SuperscriptBox[ SubscriptBox["x", "1"], "2"]}], "+", RowBox[{"4", SuperscriptBox[ SubscriptBox["y", "1"], "2"]}]}], "=", "36"}], TraditionalForm]]], ", so we can substitute the formula for the old point in terms of the new \ point ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ SubscriptBox["x", "2"], ",", SubscriptBox["y", "2"]}], ")"}], TraditionalForm]]], ". To get this formula we rotate backwards, using ", Cell[BoxData[ FormBox[ RowBox[{"-", "\[Theta]"}], TraditionalForm]]], " in place of \[Theta]. This changes the sign of ", Cell[BoxData[ FormBox[ RowBox[{"sin", "(", "\[Theta]", ")"}], TraditionalForm]]], " but not ", Cell[BoxData[ FormBox[ RowBox[{"cos", "(", "\[Theta]", ")"}], TraditionalForm]]], "." }], "Text", CellChangeTimes->{{3.42109431621875*^9, 3.4210943775*^9}, 3.4303160303977823`*^9, {3.436993162020465*^9, 3.43699324923144*^9}}], Cell[TextData[Cell[BoxData[ FormBox[GridBox[{ { RowBox[{ SubscriptBox["x", "1"], "=", RowBox[{ RowBox[{ SubscriptBox["x", "2"], RowBox[{"cos", "(", "\[Theta]", ")"}]}], "+", RowBox[{ SubscriptBox["y", "2"], RowBox[{"sin", "(", "\[Theta]", ")"}]}]}]}]}, { RowBox[{ SubscriptBox["y", "1"], "=", RowBox[{ RowBox[{ RowBox[{"-", SubscriptBox["x", "2"]}], RowBox[{"sin", "(", "\[Theta]", ")"}]}], "+", RowBox[{ SubscriptBox["y", "2"], RowBox[{"cos", "(", "\[Theta]", ")"}]}]}]}]} }], TraditionalForm]]]], "DisplayFormula", CellChangeTimes->{{3.42109438175*^9, 3.42109442740625*^9}}, GridBoxOptions->{ GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}}], Cell[TextData[{ "\tThe result is ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"9", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{ SubscriptBox["x", "2"], RowBox[{"cos", "(", "\[Theta]", ")"}]}], "+", RowBox[{ SubscriptBox["y", "2"], RowBox[{"sin", "(", "\[Theta]", ")"}]}]}], ")"}], "2"]}], "+", RowBox[{"4", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"-", SubscriptBox["x", "2"]}], RowBox[{"sin", "(", "\[Theta]", ")"}]}], "+", RowBox[{ SubscriptBox["y", "2"], RowBox[{"cos", "(", "\[Theta]", ")"}]}]}], ")"}], "2"]}]}], "=", "36"}], TraditionalForm]]], ".\n\tNow set ", Cell[BoxData[ FormBox[ RowBox[{"\[Theta]", "=", FractionBox["\[Pi]", "6"]}], TraditionalForm]]], " and multiply out." }], "Text", CellChangeTimes->{{3.421094432328125*^9, 3.421094488765625*^9}, { 3.430316034171547*^9, 3.430316039836924*^9}}], Cell[BoxData[ RowBox[{"\t", RowBox[{ RowBox[{ RowBox[{ RowBox[{"9", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"x2", " ", RowBox[{"Cos", "[", FractionBox["\[Pi]", "6"], "]"}]}], "+", RowBox[{"y2", " ", RowBox[{"Sin", "[", FractionBox["\[Pi]", "6"], "]"}]}]}], ")"}], "2"]}], "+", RowBox[{"4", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"-", "x2"}], " ", RowBox[{"Sin", "[", FractionBox["\[Pi]", "6"], "]"}]}], "+", RowBox[{"y2", " ", RowBox[{"Cos", "[", FractionBox["\[Pi]", "6"], "]"}]}]}], ")"}], "2"]}]}], "==", "36"}], " ", "//", " ", "ExpandAll"}]}]], "Input"], Cell["\tSo the equation of the new ellipse is. ", "Text", CellChangeTimes->{3.430316044107828*^9}], Cell[TextData[Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{ FractionBox["31", "4"], SuperscriptBox["x", "2"]}], "+", RowBox[{ FractionBox[ RowBox[{"5", SqrtBox["3"]}], "2"], "x", " ", "y"}], "+", RowBox[{ FractionBox["21", "4"], SuperscriptBox["y", "2"]}]}], "=", "36"}], TraditionalForm]]]], "DisplayFormula", CellChangeTimes->{{3.421094497109375*^9, 3.42109452328125*^9}}], Cell[TextData[{ "\tOnce we have the equation, we can use any variables we wish, so we go \ back to plain ", StyleBox["x", FontSlant->"Italic"], " and ", StyleBox["y", FontSlant->"Italic"], ". Let\[CloseCurlyQuote]s plot the new equation together with the old one." }], "Text", CellChangeTimes->{3.4303160464685297`*^9}], Cell[BoxData[ RowBox[{"PlotE", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"9", " ", SuperscriptBox["x", "2"]}], "+", RowBox[{"4", " ", SuperscriptBox["y", "2"]}]}], "\[Equal]", "36"}], ",", RowBox[{ RowBox[{ FractionBox[ RowBox[{"31", " ", SuperscriptBox["x", "2"]}], "4"], "+", RowBox[{ FractionBox["1", "2"], " ", RowBox[{"(", RowBox[{"5", " ", SqrtBox["3"]}], ")"}], " ", "x", " ", "y"}], "+", FractionBox[ RowBox[{"21", " ", SuperscriptBox["y", "2"]}], "4"]}], "\[Equal]", "36"}]}], "}"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "4"}], ",", "4"}], "}"}], ",", "Square"}], "]"}]], "Input", CellChangeTimes->{ 3.3918888726647387`*^9, {3.437151202775206*^9, 3.437151231174993*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Matrices and Rotations", "Section"], Cell["The Rotation Matrix", "Subsection"], Cell[TextData[{ "\tThe rotation equations can be simplified by using matrices. A ", StyleBox["matrix ", FontWeight->"Bold", FontSlant->"Italic"], "(pl. ", StyleBox["matrices", FontWeight->"Bold", FontSlant->"Italic"], ") is an array of numbers or expressions. The rotation matrix is obtained by \ taking the four coefficients from the rotation equation and arranging them in \ rows and columns like this: ", Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ { RowBox[{"cos", "(", "\[Theta]", ")"}], RowBox[{"-", RowBox[{"sin", "(", "\[Theta]", ")"}]}]}, { RowBox[{"sin", "(", "\[Theta]", ")"}], RowBox[{"cos", "(", "\[Theta]", ")"}]} }], ")"}], TraditionalForm]]] }], "Text", CellChangeTimes->{{3.421094553015625*^9, 3.42109457471875*^9}, 3.4303160504841757`*^9, {3.4369930372722263`*^9, 3.436993037853409*^9}}], Cell[TextData[{ "\tMatrices are the subject of many books and courses. Here we tell only \ enough to deal with the calculations for rotations.\n\tThe vector ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["v", FontWeight->"Bold", FontSlant->"Plain"], "=", RowBox[{"(", RowBox[{"x", ",", "y"}], ")"}]}], TraditionalForm]]], " can be written as a matrix with a single column ", Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ {"x"}, {"y"} }], ")"}], TraditionalForm]]], ", called a ", StyleBox["column vector", FontWeight->"Bold", FontSlant->"Italic"], ". You can take the dot product of a matrix with a column vector, which \ looks like this. " }], "Text", CellChangeTimes->{{3.4210945814375*^9, 3.421094625859375*^9}, { 3.430316053444626*^9, 3.430316056397195*^9}, {3.4369930460537767`*^9, 3.4369930471603413`*^9}}], Cell[TextData[Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"(", GridBox[{ { RowBox[{"cos", "(", "\[Theta]", ")"}], RowBox[{"-", RowBox[{"sin", "(", "\[Theta]", ")"}]}]}, { RowBox[{"sin", "(", "\[Theta]", ")"}], RowBox[{"cos", "(", "\[Theta]", ")"}]} }], ")"}], RowBox[{"(", GridBox[{ {"x"}, {"y"} }], ")"}]}], TraditionalForm]]]], "DisplayFormula", CellChangeTimes->{{3.421094634265625*^9, 3.421094670296875*^9}}, SpanMaxSize->Infinity], Cell[TextData[{ "\tThe meaning of this product is to take the dot product of each row of the \ matrix with the column vector, placing the answers into a new column vector, \ which is the result. 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The rotation matrix can be created using the command ", StyleBox["RotationMatrix", "Code"], ". 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(Do it!)\ \>", "Text", CellChangeTimes->{{3.437151889444091*^9, 3.43715197253473*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"Rot", "=", "RotationMatrix"}], ";"}]], "Input", CellChangeTimes->{{3.437151943444248*^9, 3.437151978576828*^9}}], Cell[TextData[{ "\tTo make a matrix of your own, just use the button on the BasicMathInput \ palette that looks like this: ", Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ {"\[Placeholder]", "\[Placeholder]"}, {"\[Placeholder]", "\[Placeholder]"} }], ")"}], TraditionalForm]]], ". Then type entries into the blanks, using the Tab key to move from one \ blank to the next. To make a column vector, select one of the columns and \ delete it." }], "Text", CellChangeTimes->{{3.4371516352324343`*^9, 3.437151743933446*^9}}], Cell[TextData[{ "\tTo rotate a point or vector, just take the dot product of the rotation \ matrix with the vector. For instance, use the following to rotate ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["v", FontWeight->"Bold", FontSlant->"Plain"], "=", RowBox[{"(", RowBox[{"1", ",", "3"}], ")"}]}], TraditionalForm]]], " by ", Cell[BoxData[ FormBox[ FractionBox["\[Pi]", "4"], TraditionalForm]]], ". 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