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    \(\(\( (*\ \(:\)\(Title : Arrow\ 3  D\)*) \)\( (*\ \(:\)\(Context : 
          Graphics`Arrow3D`\)*) \)\( (*\ \(:\)\(Author : 
          John\ M . Novak\)*) \)\( (*\ \(:\)\(Summary : 
          crude\ 3  D\ arrow\ primitive\)*) \)\( (*\ \(\(:\)\(Copyright : 
          Copyright\ 1996\ Wolfram\ Research\)\), \(\(Inc\)\(.\)\)*) \)\( (*\ \
\(:\)\(Package\ \(Version : 
            1.0\)\)*) \)\( (*\ \(:\)\(Mathematica\ \(Version : 
            3.0\)\)*) \)\( (*\ \(:\)\(History : 
          V1 \( .0--\)\ March\ 1996\ by\ John\ M . 
              Novak\)*) \)\( (*\ \(\(:\)\(Keywords : arrow\)\), 
      Graphics3D*) \)\( (*\ \(\(:\)\(Sources : 
          Tom\ Wickham - 
            Jones\)\), "\<Mathematica Graphics\>"*) \)\( (*\ \
\(\(:\)\(Discussion : 
          This\ is\ a\ package\ to\ add\ crude\ 3  D\ arrows\ by\ using\ 3  
            D\ graphics\ primitives . 
              The\ main\ problem\ with\ this\ approach\ is\ that\ arrows\ \
don' t\ have\ an\ optimal\ appearance\)\), 
      and\ distinction\ between\ \((e . g . )\)\ arrows\ point\ toward\ vs . 
          away\ from\ the\ user\ is\ difficult\ to\ make . 
          On\ the\ other\ hand, it\ is\ readily\ implementable, 
      which\ is\ far\ superior\ to\ any\ other\ approach\ available\ at\ this\
\ time . \((The\ PostScript\ used\ in\ 2  D\ arrows\ won' 
              t\ layer\ correctly\ in\ 3  D, nor\ would\ 2  D\ primitives; 
            placing\ 3  
              D\ planar\ primitives\ with\ the\ correct\ orientation\ has\ \
problems\ with\ perspective\ transforms\ as\ well\ as\ design\ problems\ with\
\ things\ like\ arrows\ with\ a\ small\ angle\ w . r . t . 
                the\ \(\(user\)\(.\)\))\)\ Because\ of\ the\ essentially\ \
different\ approach\ from\ 2  D\ arrows, 
      the\ package\ is\ being\ made\ independent\ of\ the\ 2  
        D\ package\ for\ the\ time\ \(\(being\)\(.\)\)*) \
\)\(BeginPackage["\<Graphics`Arrow3D`\>"]\[IndentingNewLine]\n
    \(Arrow3D::usage = "\<Arrow3D[start, end] is a 3D graphics primitive \
representing an arrow in
space. The start point and the end point are given as {x, y, z}
triplets.
Options controlling the length and width of the arrowhead are
available.\>";\)\n\[IndentingNewLine]
    Begin["\<`Private`\>"]\[IndentingNewLine]\[IndentingNewLine] (*The\ \
following\ routine\ is\ based\ on\ OrthogonalVectors\ from\ Tom\ Wickham - 
        Jones'\ Mathematica\ Graphics\ \(\(book\)\(.\)\)*) \n\
\[IndentingNewLine]
    anOrthogonalVector[norm : {_, _, _}] := 
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        pos = If[VectorQ[norm, NumberQ], Abs[N[norm]], 
            norm]; \[IndentingNewLine]pos = 
          Sort[Transpose[{pos, Range[3]}]]; \[IndentingNewLine]{pos, a, b} = 
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          ReplacePart[{0, 0, 0}, \(-Part[norm, a]\), 
            b]; \[IndentingNewLine]ReplacePart[v1, Part[norm, b], 
          a]]\n\[IndentingNewLine]
    normalize[vec_] := 
      vec/Sqrt[vec . 
            vec]\[IndentingNewLine]\[IndentingNewLine] (*The\ following\ \
routine' rotationmatrix'\ was\ borrowed\ from\ the\ standard\ package\ \
\(\(Graphics`SurfaceOfRevolution`\)\(.\)\)*) \n\[IndentingNewLine]
    rotationmatrix[axis_, theta_] := 
      Module[{n1, n2, n3}, {n1, n2, n3} = 
          normalize[axis] // 
            N; \[IndentingNewLine]{{n1^2 + \((1 - n1^2)\)\ Cos[theta], 
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                n3\ Sin[theta], n2^2 + \((1 - n2^2)\)\ Cos[theta], 
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                n1\ Sin[theta]}, {n1\ n3\ \((1 - Cos[theta])\) + 
                n2\ Sin[theta], 
              n2\ n3\ \((1 - Cos[theta])\) - n1\ Sin[theta], 
              n3^2 + \((1 - n3^2)\)\ Cos[theta]}} // N]\n\[IndentingNewLine]
    Arrow3D[base : {_, _, _}, tip : {_, _, _}, polys_:  10, 
        len_:  0.8] := {Line[{base, tip}], 
        mycone[tip - base, base, tip, polys, len]}\n\[IndentingNewLine]
    Arrow3D[base : {_, _, _}, tip : {_, _, _}, ___] := 
      Point[base] /; base \[Equal] tip\n\[IndentingNewLine]
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      Block[{norm = anOrthogonalVector[vec], theta}, 
        Map[Polygon[Append[#, tip]] &, 
          Partition[
            Table[base + 
                  len\ vec + \((1 - len)\)/2\ norm . 
                      rotationmatrix[vec, theta], {theta, 0, 2\ Pi, 
                  2\ Pi/polys}] // N, 2, 1]]]\n\[IndentingNewLine]
    End[]\n\[IndentingNewLine]
    EndPackage[]\n\[IndentingNewLine]\n
    \("\<Graphics`Arrow3D`\>";\)\n
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in space. The start point and the end point are given as {x, y, z} triplets. \
Options controlling the length and width of the arrowhead are \
available.\>";\)\n
    \("\<Graphics`Arrow3D`Private`\>";\)\n
    \("\<Graphics`Arrow3D`Private`\>";\)\[IndentingNewLine]
    \)\)\)], "Input",
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Cell[TextData[{
  Cell[BoxData[{
      StyleBox[\(MATH\ 257\ Calculus\ III\t\t\t\t\tWeek\ of\ May\ 5, \ 2003\),
        
        FontSize->14], "\n", 
      StyleBox[\(Lab\ 6 : \ Wectors\),
        "Title"], "\n", 
      StyleBox[\(Name\ 1\),
        "Section"], "\n", 
      StyleBox[\(Name\ 2\),
        "Section"], "\n", 
      StyleBox[\(\(Section\)\(:\)\),
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  "\n"
}], "Text",
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Cell[TextData[{
  StyleBox["Before you begin",
    FontWeight->"Bold"],
  ", execute the following command.  YOU WILL BE ASKED if you want to execute \
any initialization cells.  CLICK YES."
}], "Text",
  FontSize->16],

Cell[BoxData[
    RowBox[{
      StyleBox[\(<< Graphics`Arrow`;\),
        "Input"], "\[IndentingNewLine]"}]], "Input"],

Cell[TextData[{
  "I.  Laboratory Objectives\n\tUse ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " to explore\n\t\t3D graphics and Vectors.  Yeah!"
}], "Text"],

Cell[TextData[StyleBox["II.  Formatting and Syntax Information", "Text"]], \
"Text"],

Cell[TextData[{
  "\t",
  StyleBox["Show[Graphics[ ]", "Input"],
  " \tThese make ",
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  " render whatever\n\t",
  StyleBox["Show[Graphics3D[ ]]\t", "Input"],
  "\t2D or 3D graphics you've conconcted (which\n\t\t\t\t\tyou have to put in \
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  StyleBox["Arrow[{x1,y1}, {x2,y2}]", "Input"],
  "\tDraws an Arrow from {x1,y1} to {x2,y2}.\n\t   \t\t\t\t\tYou need to do \
",
  StyleBox["<<Graphics`Arrow`", "Input"],
  " first\n\t   \t\t\t\t\tto load the Arrow package.\n\t",
  StyleBox["Arrow3D[ pt1, pt2 ]", "Input"],
  "\tSame thing as Arrow, but for 3D vectors.  Here\n\t\t\t\t\tpt1 must have \
three coordinates, {x1,y1,z1}.  Ditto\n\t\t\t\t\tfor pt2.  This is NOT a \
standard ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " command.\n\t\t\t\t\tBut the secret initialization cells in this lab will \
define it...\n\t\t\t\t\t\n\t",
  StyleBox["v.u", "Input"],
  "\t\t\t\tThe ",
  StyleBox["Dot Product",
    FontWeight->"Bold"],
  " of two vectors ",
  StyleBox["v", "Input"],
  " and ",
  StyleBox["u", "Input"],
  ".  \n\t\t\t\t\tDon't know what this is?  You'll see below...\n\t\t\t\t\t\n\
"
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Cell[TextData[StyleBox["Vectors (or Wectors) in 2D",
  FontWeight->"Bold",
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Cell[TextData[{
  StyleBox["In class (and in the book), we denote a vector like this:  v = \
<1, 3>.  In ", "Text",
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    FontSize->14,
    FontWeight->"Plain"],
  StyleBox["Mathematica", "Text",
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    FontWeight->"Plain",
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  StyleBox[", we express vectors as LISTS:", "Text",
    FontFamily->"Times",
    FontSize->14,
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  "If we wanted to draw this vector centered at the origin, we can use the \
\"Arrow\" command.  (If this does NOT work, you didn't do the ",
  StyleBox["<<Graphics`Arrow`", "Input"],
  " command above.  Go do it!)"
}], "Text",
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      Graphics[{Thickness[ .01], Arrow[{0, 0}, v]}]]\)\)\)], "Input"],

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OK.  That doesn't picture doesn't say much.  Try it again, this \
time adding axes, and making them scaled the same:
\
\>", "Text",
  FontSize->14],

Cell[BoxData[
    \(Show[Graphics[{Thickness[ .01], Arrow[{0, 0}, v]}], \ 
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  "Therefore, the ",
  StyleBox["Arrow[{x1, y1}, {x2, y2}]", "Input"],
  " command will draw an arrow ",
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    FontWeight->"Bold"],
  " at (x1,y1) and ",
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  " at (x2,y2)."
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Cell[TextData[{
  "\n",
  StyleBox["Question 1:",
    FontWeight->"Bold",
    FontColor->RGBColor[1, 0, 0]],
  StyleBox[" Make a picture of 5 vectors that form a nice pentagon.  (It \
doesn't have to be perfectly regular, but try your best.)",
    FontColor->RGBColor[1, 0, 0]]
}], "Text",
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Cell[TextData[StyleBox[" Vectors in 3D",
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  "\nTo make vectors in 3D space, we need to use the ",
  StyleBox["Arrow3D[ ]", "Input"],
  " command. This is NOT a standard ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " command; it was loaded when you clicked \"yes\" to the initialization \
cells.  Clever, eh?\n\nLet's define a 3D vector and draw it centered at the \
origin."
}], "Text",
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Cute, eh?  
\
\>", "Text",
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Cell[TextData[{
  StyleBox["Question 2a:",
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  " Make a picture of a bunch of vectors that form a perfect ",
  StyleBox["cube",
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  ".  This is FUN!  ",
  StyleBox["Both",
    FontWeight->"Bold"],
  " people in your group should do this to get the experience.  (Race to see \
who gets it first.)  \n",
  StyleBox["Do it this way:",
    FontWeight->"Bold"],
  "  let one corner of the cube be at {0,0,0}, and let the opposite corner be \
{1,1,1}.  Let ",
  StyleBox["o={0,0,0}", "Input"],
  " be the origin, and label your other corners ",
  StyleBox["c1, c2, ..., c7", "Input"],
  ".  That way you won't have to keep typing them in.  Here, I'll get you \
started..."
}], "Text",
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Cell[TextData[{
  StyleBox["Question 2b:",
    FontWeight->"Bold"],
  " Now that you've done question 2a, do the below command and ",
  StyleBox["explain",
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  " what you see in a sentance or two:"
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Cell[TextData[StyleBox[" Vectors do Work",
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  FontColor->RGBColor[0, 0, 1]]], "Subtitle"],

Cell[TextData[{
  "(READ THIS.  You're gonna learn something! (Though you may have seen this \
before in a physics class...))\nOne of the big uses of vectors is to solve \
physics Work problems that are more complicated than the ones you saw last \
quarter.  Recall that the Work done when applying a Force over a Distance is:\
\n\t\t\t\tWork = Force * Distance\nThis assumes that the force is acting in \
the ",
  StyleBox["same direction",
    FontWeight->"Bold"],
  " as the distance being travelled.  remember that.\n\nSo suppose we have a \
heavy box sitting at the origin in the x-y plane.  We tie a rope to it and, \
while standing on the x-axis, pull on the rope with a force of 20 Newtons.  \
The situation can be viewed as in the below picture:"
}], "Text",
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",
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