(* 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[ 26206, 731] NotebookOptionsPosition[ 22589, 628] NotebookOutlinePosition[ 22992, 645] CellTagsIndexPosition[ 22949, 642] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Lecture 6", "Section", CellChangeTimes->{{3.426947592244849*^9, 3.4269475961628494`*^9}, { 3.4315423019150314`*^9, 3.4315423352420316`*^9}, {3.4327427093176003`*^9, 3.4327427098915997`*^9}, {3.432826830388052*^9, 3.432826830967052*^9}}], Cell["\<\ In the Lecture we discuss the use of ordered lists of numbers, either lists \ with 1 index (vectors), or 2 indices (matrices). These constructs are \ extremely important in computational work and Mathematica offers many tools \ to deal with them. In this notebook we will introduce some of those tools. \ Since Mathematica is ready to deal with lists of numbers, i.e., components of \ vectors, we should assume that the decision about the definition of the \ specific basis vectors has been made. A vector is entered directly as a \ \"list\" (of components), either explicitly (note the curly brackets) or with \ the List command,\ \>", "Text", CellChangeTimes->{{3.4269476003598495`*^9, 3.4269476357228494`*^9}, 3.4315423175140314`*^9, {3.431542352626032*^9, 3.431542424331032*^9}, { 3.4327427140746*^9, 3.4327429812265997`*^9}, 3.4327511315046*^9, 3.4327511653416*^9, {3.432826858714052*^9, 3.432827111073052*^9}, { 3.432827229514052*^9, 3.4328272577850523`*^9}, {3.432827310610052*^9, 3.432827312426052*^9}, 3.432827345393052*^9, {3.4328274568730516`*^9, 3.432827466097052*^9}, {3.4328340733070517`*^9, 3.432834109282052*^9}}], Cell[BoxData[ RowBox[{"v", "=", RowBox[{"{", RowBox[{"x", ",", "y", ",", "z"}], "}"}]}]], "Input", CellChangeTimes->{{3.4315425582480316`*^9, 3.4315425645630317`*^9}, { 3.431543892739032*^9, 3.4315439015950317`*^9}, {3.4315440829790316`*^9, 3.431544083211032*^9}, {3.4315472267070312`*^9, 3.4315472270590315`*^9}, { 3.4327432057306004`*^9, 3.4327432879056*^9}, {3.4327447790456*^9, 3.4327447994986*^9}, {3.4328273021790524`*^9, 3.432827304691052*^9}, { 3.4328273870430517`*^9, 3.432827397018052*^9}}], Cell[BoxData["v"], "Input", CellChangeTimes->{3.432827400872052*^9}], Cell["Or", "Text", CellChangeTimes->{{3.432827406006052*^9, 3.4328274096490517`*^9}}], Cell[BoxData[ RowBox[{"v", "=", RowBox[{"List", "[", RowBox[{"x", ",", "y", ",", "z"}], "]"}]}]], "Input", CellChangeTimes->{{3.432827412861052*^9, 3.432827438138052*^9}}], Cell[BoxData[ RowBox[{"MatrixForm", "[", "v", "]"}]], "Input", CellChangeTimes->{{3.432834349494052*^9, 3.432834353683052*^9}}], Cell["\<\ We select individual components using double square brackets -\ \>", "Text", CellChangeTimes->{{3.432827475390052*^9, 3.432827498105052*^9}}], Cell[BoxData[ RowBox[{"v", "[", RowBox[{"[", "2", "]"}], "]"}]], "Input", CellChangeTimes->{{3.432827505157052*^9, 3.432827507119052*^9}}], Cell["\<\ To find the length or norm of the vector use Norm (note that, as suggested in \ the notes, Mathematica assumes the possibility of complex valued components, \ see Eq. 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In the Euclidean case\ \>", "Text", CellChangeTimes->{{3.4327452715506*^9, 3.4327452836546*^9}, { 3.4328292661230516`*^9, 3.432829297610052*^9}}], Cell[BoxData[ RowBox[{"g", "=", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"1", ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "1", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "0", ",", "1"}], "}"}]}], "}"}]}]], "Input", CellChangeTimes->{{3.4327452875025997`*^9, 3.4327453072066*^9}, { 3.432829302347052*^9, 3.4328293310500517`*^9}}], Cell["Or more familiarly", "Text", CellChangeTimes->{{3.432829335721052*^9, 3.432829362265052*^9}}], Cell[BoxData[ RowBox[{"MatrixForm", "[", "g", "]"}]], "Input", CellChangeTimes->{{3.4315425582480316`*^9, 3.4315425645630317`*^9}, { 3.431543892739032*^9, 3.4315439015950317`*^9}, {3.4315440829790316`*^9, 3.431544083211032*^9}, {3.4315472267070312`*^9, 3.4315472270590315`*^9}, { 3.4327432057306004`*^9, 3.4327432879056*^9}, {3.4327438702706003`*^9, 3.4327438961706*^9}, {3.4327448646636*^9, 3.4327448738106003`*^9}, 3.4327453271176*^9, {3.432829366619052*^9, 3.432829378171052*^9}}], Cell["With scalar product defined by (see Eq. (6.14))", "Text", CellChangeTimes->{{3.4327454490446*^9, 3.4327454597706003`*^9}, { 3.4327455344256*^9, 3.4327455452995996`*^9}, {3.4327512651116*^9, 3.4327512784846*^9}, {3.432829388386052*^9, 3.4328294424170523`*^9}}], Cell[BoxData[ RowBox[{"v1", ".", "g", ".", "v2"}]], "Input", CellChangeTimes->{{3.4327453624486*^9, 3.4327454292186003`*^9}, { 3.4327455592216*^9, 3.4327456018486*^9}, {3.432829407723052*^9, 3.432829420394052*^9}}], Cell[BoxData[ RowBox[{"v1", ".", "v2"}]], "Input", CellChangeTimes->{{3.432829453959052*^9, 3.432829455850052*^9}}], Cell["\<\ Note that we use 2 indices to select a specific element in a matrix\ \>", "Text", CellChangeTimes->{{3.4328341915350523`*^9, 3.432834214433052*^9}}], Cell[BoxData[ RowBox[{"g", "[", RowBox[{"[", RowBox[{"3", ",", "3"}], "]"}], "]"}]], "Input", CellChangeTimes->{{3.432834216992052*^9, 3.4328342234500523`*^9}}], Cell["or 1 for a row", "Text", CellChangeTimes->{{3.432834237830052*^9, 3.432834241393052*^9}}], Cell[BoxData[ RowBox[{"g", "[", RowBox[{"[", "2", "]"}], "]"}]], "Input", CellChangeTimes->{{3.432834244408052*^9, 3.4328342462480516`*^9}}], Cell["Now consider the vector or cross product provided by", "Text", CellChangeTimes->{{3.4327456568386*^9, 3.4327456716126003`*^9}, { 3.4327458102276*^9, 3.4327458265126*^9}, {3.4327471856296*^9, 3.4327472068766003`*^9}, {3.4327512909765997`*^9, 3.4327513216426*^9}, { 3.432829467732052*^9, 3.432829488497052*^9}, {3.432834263898052*^9, 3.432834280689052*^9}}], Cell[BoxData[ RowBox[{"v3", "=", RowBox[{"Cross", "[", RowBox[{"v1", ",", "v2"}], "]"}]}]], "Input", CellChangeTimes->{{3.4315425582480316`*^9, 3.4315425645630317`*^9}, { 3.431543892739032*^9, 3.4315439015950317`*^9}, {3.4315440829790316`*^9, 3.431544083211032*^9}, {3.4315472267070312`*^9, 3.4315472270590315`*^9}, { 3.4327432057306004`*^9, 3.4327432879056*^9}, {3.4327447790456*^9, 3.4327447994986*^9}, {3.4327457223556004`*^9, 3.4327457595956*^9}, { 3.4327458430366*^9, 3.4327458934695997`*^9}, {3.4327459714596*^9, 3.4327459729726*^9}, {3.432829491652052*^9, 3.432829510243052*^9}}], Cell["Orthogonal to both", "Text", CellChangeTimes->{{3.432829549415052*^9, 3.432829553313052*^9}}], Cell[BoxData[ RowBox[{"Simplify", "[", RowBox[{"v3", ".", "v1"}], "]"}]], "Input", CellChangeTimes->{{3.432829517064052*^9, 3.432829534339052*^9}}], Cell[BoxData[ RowBox[{"Simplify", "[", RowBox[{"v3", ".", "v2"}], "]"}]], "Input", CellChangeTimes->{{3.4328295372080517`*^9, 3.432829542754052*^9}}], Cell["Not commutative", "Text", CellChangeTimes->{{3.432829585427052*^9, 3.432829589433052*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"Cross", "[", RowBox[{"v1", ",", "v2"}], "]"}], "-", RowBox[{"Cross", "[", RowBox[{"v2", ",", "v1"}], "]"}]}]], "Input", CellChangeTimes->{{3.432829593494052*^9, 3.432829608658052*^9}}], Cell["But associative", "Text", CellChangeTimes->{{3.432829632838052*^9, 3.432829642321052*^9}}], Cell[BoxData[ RowBox[{"Simplify", "[", RowBox[{ RowBox[{"v", ".", RowBox[{"Cross", "[", RowBox[{"v1", ",", "v2"}], "]"}]}], "-", RowBox[{ RowBox[{"Cross", "[", RowBox[{"v", ",", "v1"}], "]"}], ".", "v2"}]}], "]"}]], "Input", CellChangeTimes->{{3.432829648654052*^9, 3.432829697850052*^9}}], Cell["And distributive", "Text", CellChangeTimes->{{3.4328297186230516`*^9, 3.432829727265052*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"Cross", "[", RowBox[{"v", ",", RowBox[{"v1", "+", "v2"}]}], "]"}], "-", RowBox[{"Cross", "[", RowBox[{"v", ",", "v1"}], "]"}], "-", RowBox[{"Cross", "[", RowBox[{"v", ",", "v2"}], "]"}]}]], "Input", CellChangeTimes->{{3.432829735206052*^9, 3.432829769066052*^9}, { 3.432829852659052*^9, 3.4328298587780523`*^9}}], Cell["\<\ Now consider the explicit examples following Eq. (6.35) with points (vectors)\ \ \>", "Text", CellChangeTimes->{{3.4327459884026003`*^9, 3.4327459914835997`*^9}, { 3.4327461126146*^9, 3.4327461347066*^9}, {3.432830052611052*^9, 3.432830093761052*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"r1", "=", RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", "1", ",", "1"}], "}"}]}], ";", RowBox[{"r2", "=", RowBox[{"{", RowBox[{"2", ",", "3", ",", "0"}], "}"}]}], ";", RowBox[{"r3", "=", RowBox[{"{", RowBox[{"0", ",", "1", ",", RowBox[{"-", "2"}]}], "}"}]}], ";", " "}]], "Input", CellChangeTimes->{{3.4327459995106*^9, 3.4327460093316*^9}, { 3.4327461380256*^9, 3.4327461717016*^9}, {3.4327462094806004`*^9, 3.4327462103096*^9}, 3.4327513719336*^9, {3.4328300963730516`*^9, 3.4328301452190523`*^9}}], Cell["Defining a normal, Eq. (6.36)", "Text", CellChangeTimes->{{3.4327462199126*^9, 3.4327462226776*^9}, { 3.432830152249052*^9, 3.4328301774170523`*^9}}], Cell[BoxData[ RowBox[{"n", "=", RowBox[{"Cross", "[", RowBox[{ RowBox[{"r2", "-", "r1"}], ",", RowBox[{"r3", "-", "r1"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.4327459995106*^9, 3.4327460093316*^9}, { 3.4327461380256*^9, 3.4327461717016*^9}, {3.432830185546052*^9, 3.432830205402052*^9}}], Cell["Thus the equation for the plane is given by", "Text", CellChangeTimes->{{3.4327463833606*^9, 3.4327465234526*^9}, { 3.4327471339365997`*^9, 3.4327471506675997`*^9}, {3.4327475393426*^9, 3.4327475751036*^9}, {3.4327513472106*^9, 3.4327513772086*^9}, { 3.432830224652052*^9, 3.4328302508250523`*^9}}], Cell[BoxData[ RowBox[{"sol", "=", RowBox[{"Solve", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"v", "-", "r1"}], ")"}], ".", "n"}], "\[Equal]", "0"}], ",", RowBox[{"{", RowBox[{"x", ",", "y", ",", "z"}], "}"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.4327470776156*^9, 3.4327471600216*^9}, { 3.4328302621790524`*^9, 3.432830309802052*^9}, {3.4328303828590517`*^9, 3.432830385363052*^9}}], Cell[BoxData[ RowBox[{"x", "=", RowBox[{"x", "/.", RowBox[{"sol", "[", RowBox[{"[", "1", "]"}], "]"}]}]}]], "Input", CellChangeTimes->{{3.4328303929360523`*^9, 3.432830414363052*^9}}], Cell["Which is Eq. (6.37), 3x - 4y + z + 6 = 0.", "Text", CellChangeTimes->{{3.432830329138052*^9, 3.4328303714170523`*^9}, 3.432834034758052*^9}], Cell["Next consider a line represented by ", "Text", CellChangeTimes->{{3.4328327699430523`*^9, 3.4328327878250523`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"r0", "=", RowBox[{"{", RowBox[{"1", ",", "2", ",", "0"}], "}"}]}], ";", RowBox[{"v0", "=", RowBox[{"{", RowBox[{"2", ",", "0", ",", RowBox[{"-", "1"}]}], "}"}]}], ";", RowBox[{ RowBox[{"r", "[", "t_", "]"}], ":=", RowBox[{"r0", "+", RowBox[{"v0", " ", "t"}]}]}]}]], "Input", CellChangeTimes->{{3.4328328325490522`*^9, 3.432832896987052*^9}, { 3.432832965963052*^9, 3.432832983843052*^9}, 3.4328330771410522`*^9}], Cell[BoxData[ RowBox[{"r", "[", "t", "]"}]], "Input", CellChangeTimes->{{3.4328329044250517`*^9, 3.432832906213052*^9}}], Cell["The minimum distance from the following point", "Text", CellChangeTimes->{{3.432833088079052*^9, 3.4328330988250523`*^9}, { 3.432833163282052*^9, 3.4328331662010517`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"r1", "=", RowBox[{"{", RowBox[{"1", ",", "1", ",", "1"}], "}"}]}], ";"}]], "Input"], Cell["to the line is", "Text", CellChangeTimes->{{3.432833103030052*^9, 3.432833105105052*^9}}], Cell[BoxData[ RowBox[{"d", "=", RowBox[{ RowBox[{"Norm", "[", RowBox[{"Cross", "[", RowBox[{ RowBox[{"r1", "-", RowBox[{"r", "[", "t", "]"}]}], ",", "v0"}], "]"}], "]"}], "/", RowBox[{"Norm", "[", "v0", "]"}]}]}]], "Input", CellChangeTimes->{{3.432832938088052*^9, 3.4328329545540524`*^9}, { 3.432832993051052*^9, 3.432833044155052*^9}}], Cell["\<\ The minimum distance from the same point to the plane with the normal vector \ n above and through the point at {0,0,-6} is given by \ \>", "Text", CellChangeTimes->{{3.432833153928052*^9, 3.432833225481052*^9}, 3.432834045398052*^9}], Cell[BoxData["n"], "Input", CellChangeTimes->{3.4328333200420523`*^9}], Cell[BoxData[ RowBox[{"r0", "=", RowBox[{"{", RowBox[{"0", ",", "0", ",", "6"}], "}"}]}]], "Input", CellChangeTimes->{{3.432833233658052*^9, 3.432833239306052*^9}}], Cell[BoxData[ RowBox[{"d2", "=", RowBox[{ RowBox[{"Abs", "[", RowBox[{ RowBox[{"(", RowBox[{"r1", "-", "r0"}], ")"}], ".", "n"}], "]"}], "/", RowBox[{"Norm", "[", "n", "]"}]}]}]], "Input", CellChangeTimes->{{3.432833248291052*^9, 3.432833314955052*^9}}], Cell["which is the result of Eq. (6.41)", "Text", CellChangeTimes->{{3.432833348954052*^9, 3.432833386473052*^9}}], Cell["Finally consider the intersecting 2 plane problem.", "Text", CellChangeTimes->{{3.4327473616506*^9, 3.4327474691516*^9}, { 3.4327477693898*^9, 3.4327477888758*^9}, {3.4327478309378*^9, 3.4327479603478003`*^9}, {3.4327514019016*^9, 3.4327514225425997`*^9}, { 3.432833450466052*^9, 3.432833494065052*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"n1", "=", RowBox[{"{", RowBox[{"1", ",", RowBox[{"-", "2"}], ",", "3"}], "}"}]}], ";", RowBox[{"n2", "=", RowBox[{"{", RowBox[{"2", ",", "1", ",", RowBox[{"-", "1"}]}], "}"}]}], ";"}]], "Input", CellChangeTimes->{{3.4327453624486*^9, 3.4327454292186003`*^9}, { 3.4327455592216*^9, 3.4327456018486*^9}, {3.4327472814716*^9, 3.4327473417855997`*^9}, {3.4327479735278*^9, 3.4327480032518*^9}, { 3.4327480638118*^9, 3.4327480651988*^9}, {3.432833503192052*^9, 3.432833547475052*^9}}], Cell[BoxData[ RowBox[{"v", "=", RowBox[{"Cross", "[", RowBox[{"n1", ",", "n2"}], "]"}]}]], "Input", CellChangeTimes->{{3.432833559073052*^9, 3.4328335741790524`*^9}}], Cell["\<\ Finding a point on both planes with the choice z = 0 we have (clearing \ previous definitions first)\ \>", "Text", CellChangeTimes->{{3.4328335872660522`*^9, 3.432833609489052*^9}, { 3.432833763034052*^9, 3.432833775809052*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"Clear", "[", "x", "]"}], ";", RowBox[{"Clear", "[", "y", "]"}], ";", RowBox[{"Clear", "[", "z", "]"}], ";"}]], "Input", CellChangeTimes->{{3.4328337344180517`*^9, 3.4328337528580523`*^9}}], Cell[BoxData[ RowBox[{"sol", "=", RowBox[{"Solve", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"x", " ", "-", " ", RowBox[{"2", "y"}], " ", "+", RowBox[{"3", " ", "z"}]}], "\[Equal]", "4"}], ",", RowBox[{ RowBox[{ RowBox[{"2", " ", "x"}], "+", "y", " ", "-", " ", "z"}], " ", "\[Equal]", "5"}], ",", RowBox[{"z", " ", "\[Equal]", " ", "0"}]}], "}"}], ",", RowBox[{"{", RowBox[{"x", ",", "y", ",", "z"}], "}"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.432833612004052*^9, 3.432833660923052*^9}, { 3.4328338376670523`*^9, 3.432833838699052*^9}}], Cell["Select the values", "Text", CellChangeTimes->{{3.432834781558052*^9, 3.4328347849690523`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"x", "=", RowBox[{"x", "/.", RowBox[{"sol", "[", RowBox[{"[", RowBox[{"1", ",", "1"}], "]"}], "]"}]}]}], ";", RowBox[{"y", "=", RowBox[{"y", "/.", RowBox[{"sol", "[", RowBox[{"[", RowBox[{"1", ",", "2"}], "]"}], "]"}]}]}], ";", RowBox[{"z", "=", RowBox[{"z", "/.", RowBox[{"sol", "[", RowBox[{"[", RowBox[{"1", ",", "3"}], "]"}], "]"}]}]}], ";"}]], "Input", CellChangeTimes->{{3.432833842573052*^9, 3.4328339228190517`*^9}}], Cell["and so ", "Text", CellChangeTimes->{{3.4327480118918*^9, 3.4327480146678*^9}, { 3.432833791490052*^9, 3.432833793489052*^9}}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"r", "[", "t_", "]"}], ":=", RowBox[{ RowBox[{"{", RowBox[{"x", ",", "y", ",", "z"}], "}"}], "+", RowBox[{"v", " ", "t"}]}]}], ";", RowBox[{"r", "[", "t", "]"}]}]], "Input", CellChangeTimes->{{3.4327453624486*^9, 3.4327454292186003`*^9}, { 3.4327455592216*^9, 3.4327456018486*^9}, {3.4327472814716*^9, 3.4327473417855997`*^9}, {3.4327480374688*^9, 3.4327480540758*^9}, { 3.432833803820052*^9, 3.432833817468052*^9}, {3.4328339294760523`*^9, 3.4328339596820517`*^9}}], Cell["which is Eq. 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