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          crude\ 3  D\ arrow\ primitive\)*) \)\( (*\ \(\(:\)\(Copyright : 
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\(\(:\)\(Discussion : 
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don' t\ have\ an\ optimal\ appearance\)\), 
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      StyleBox[\(MATH\ 257\ Calculus\ III\t\t\t\tWeek\ of\ May\ 12, \ 2003\),
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      StyleBox[\(Lab\ 7 : \ Cross\ Products\ and\ Planes\),
        "Title"], "\n", 
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  "I.  Laboratory Objectives\n\tUse ",
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  " vectors, and matrices to explore\n\t\tCross Products"
}], "Text"],

Cell[TextData[StyleBox["II.  Formatting and Syntax Information", "Text"]], \
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  "\t\t\t\tTwo ways to do the ",
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Cell[TextData[{
  StyleBox["Question 3:",
    FontWeight->"Bold"],
  " Suppose you're given three points of a triangle: (0,0,0), (x1, y1, 0), \
and (x2, y2, 0). Use the cross product to generate a formula (in terms of the \
x1, x2, y1, and y2) of the area of this triangle.  (",
  StyleBox["Hint:",
    FontWeight->"Bold"],
  " think half of a parallelogram.)\n"
}], "Text",
  FontSize->14,
  FontColor->RGBColor[1, 0, 0]],

Cell[BoxData[
    \(\[IndentingNewLine]\)], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData[StyleBox["Graphing Planes in 3D",
  FontWeight->"Bold",
  FontColor->RGBColor[0, 0, 1]]], "Subtitle"],

Cell[TextData[{
  "Graphing 3D surfaces in ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " is easy if you have a function of the form z=f(x,y).  Just use the ",
  StyleBox["Plot3D[]", "Input"],
  " command, which works just like ",
  StyleBox["Plot[]", "Input"],
  " except you have to specify the x ",
  StyleBox["and",
    FontWeight->"Bold"],
  " y ranges.\n\nFor example, let's plot the plane  ",
  StyleBox["z = -x +.25 y +1",
    FontWeight->"Bold"],
  ".   (Here we use the ",
  StyleBox["ViewPoint", "Input"],
  " command to get a better view of it.)"
}], "Text",
  FontSize->14],

Cell[BoxData[
    \(\(\(\[IndentingNewLine]\)\(plane1 = 
      Plot3D[\(-x\)\  + \  .25*y\  + \ 1, \ {x, \ \(-2\), 2}, \ {y, \(-2\), 
          2}, \ PlotRange \[Rule] {\(-2\), 2}, \ AspectRatio \[Rule] 1, \ 
        ViewPoint -> {2.318, \ \(-2.360\), \ 0.713}]\)\)\)], "Input"],

Cell[TextData[{
  "(",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " chops off parts of the plane that don't fit.  That's what you see at the \
top.)\nLet's draw some vectors on this plane.  Plugging in",
  StyleBox[" x=0", "Input"],
  ", ",
  StyleBox["y=0", "Input"],
  ", we get ",
  StyleBox["z=1", "Input"],
  ", so the point (0,0,1) is on this plane. We can also plug in ",
  StyleBox["x=1", "Input"],
  ", ",
  StyleBox["y=0", "Input"],
  " and plug in ",
  StyleBox["x=0", "Input"],
  ", ",
  StyleBox["y=1", "Input"],
  " to get other points.  Then we can draw some arrows connecting these \
points:\n"
}], "Text",
  FontSize->14],

Cell[BoxData[{
    \(\(p = {0, 0, 1};\)\), "\[IndentingNewLine]", 
    \(\(p1 = {1, 0, 0};\)\), "\[IndentingNewLine]", 
    \(\(p2 = {0, 1, 1.25};\)\), "\[IndentingNewLine]", 
    \(Show[{plane1, \ 
        Graphics3D[{Thickness[ .01], Arrow3D[p, p1], \ 
            Arrow3D[p, p2]}]}]\)}], "Input"],

Cell[TextData[{
  "Notice how we used the ",
  StyleBox["Show[]", "Input"],
  " command to combine the plane1 plot with the ",
  StyleBox["Graphics3D", "Input"],
  " arrow commands.  \n\nSo, let's find the cross product of these two \
vectors!  We ",
  StyleBox["cannot",
    FontWeight->"Bold"],
  " just use",
  StyleBox[" p", "Input"],
  ", ",
  StyleBox["p1", "Input"],
  ", and ",
  StyleBox["p2", "Input"],
  " as they are, since they would be vectors going from the ",
  StyleBox["origin",
    FontWeight->"Bold"],
  " to these points.  Instead, we gotta subtract them:  one vector will be ",
  StyleBox["p1-p", "Input"],
  ", and the other will be ",
  StyleBox["p2-p", "Input"],
  ".  So call their cross product ",
  StyleBox["Nvec", "Input"],
  " (since it will be a normal vector), and let's graph them all together!",
  "\n"
}], "Text",
  FontSize->14],

Cell[BoxData[{
    \(Nvec = \((p1 - p)\)\[Cross]\((p2 - p)\)\), "\[IndentingNewLine]", 
    \(Show[{plane1, \ 
        Graphics3D[{Thickness[ .01], Arrow3D[p, p1], \ 
            Arrow3D[p, p2], \[IndentingNewLine]Arrow3D[p, 
              Nvec + p]}]}]\)}], "Input"],

Cell[TextData[{
  "Notice how the perspective is a little weird, but it does look like ",
  StyleBox["Nvec", "Input"],
  " is normal to the plane!"
}], "Text",
  FontSize->14],

Cell[TextData[{
  StyleBox["Question 4:",
    FontWeight->"Bold"],
  " Use the same procedure we used above to make a picture of the plane \
z=3x+y together with a normal vector at the point (0,0,0)."
}], "Text",
  FontSize->14,
  FontColor->RGBColor[1, 0, 0]],

Cell[BoxData[
    \(\[IndentingNewLine]\)], "Input"],

Cell[TextData[{
  "Here's how you can make ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " plot these two planes at the same time:"
}], "Text",
  FontSize->14],

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