View of /trunk/NationalProblemLibrary/WHFreeman/Rogawski_Calculus_Early_Transcendentals_Second_Edition/12_Vector_Geometry/12.3_Dot_Product_and_the_Angle_Between_Two_Vectors/12.3.11.pg

    1 ## DBsubject('Calculus')
    2 ## DBchapter('Vector Geometry')
    3 ## DBsection('Dot Product and the Angle Between Two Vectors')
    4 ## KEYWORDS('calculus', 'parametric', 'vector', 'dot product', 'scalar product', 'angle', 'projection', 'proj')
    5 ## TitleText1('Calculus: Early Transcendentals')
    6 ## EditionText1('2')
    7 ## AuthorText1('Rogawski')
    8 ## Section1('12.3')
    9 ## Problem1('11')
   10 ## Author('Christopher Sira')
   11 ## Institution('W.H.Freeman')
   12 
   13 DOCUMENT();
   14 loadMacros("PG.pl","PGbasicmacros.pl","PGanswermacros.pl");
   15 loadMacros("PGchoicemacros.pl");
   16 loadMacros("Parser.pl");
   17 loadMacros("freemanMacros.pl");
   18 $context = Context("Vector");
   19 Context()->flags->set(ijk=>1);
   20 
   21 sub vdot
   22 {
   23     my($temp) = 0;
   24 
   25     for ($i = 0; $i < 3; $i++)
   26     {
   27         $temp = $temp + $_[$i] * $_[$i + 3];
   28     }
   29     return $temp
   30 }
   31 
   32 sub vlen
   33 {
   34     return sqrt($_[0]**2 + $_[1]**2 + $_[2]**2);
   35 }
   36 
   37 sub vcos
   38 {
   39     my($dot) = vdot(@_);
   40 
   41     return $dot / (vlen($_[0], $_[1], $_[2]) * vlen($_[3], $_[4], $_[5]));
   42 }
   43 
   44 sub vang
   45 {
   46     return Formula("arccos(x)")->eval(x=>vcos(@_));
   47 }
   48 
   49 # projection of u onto v (first onto second)
   50 sub vproj
   51 {
   52     my(@temp) = ($_[3], $_[4], $_[5]);
   53     my($temp2) = vdot(@_);
   54 
   55     $temp2 = $temp2 / vdot($_[3], $_[4], $_[5], $_[3], $_[4], $_[5]);
   56 
   57     $temp[0] *= $temp2;
   58     $temp[1] *= $temp2;
   59     $temp[2] *= $temp2;
   60 
   61     return @temp;
   62 }
   63 
   64 # perpendicular component of u onto v (first onto second)
   65 sub vperp
   66 {
   67     my(@temp) = vproj(@_);
   68 
   69     $temp[0] = -1 * $temp[0] + $_[0];
   70     $temp[1] = -1 * $temp[1] + $_[1];
   71     $temp[2] = -1 * $temp[2] + $_[2];
   72 
   73     return @temp;
   74 }
   75 
   76 $iv0 = 1;
   77 $iv1 = 0;
   78 $iv2 = 0;
   79 
   80 $jv0 = 0;
   81 $jv1 = 1;
   82 $jv2 = 0;
   83 
   84 $kv0 = 0;
   85 $kv1 = 0;
   86 $kv2 = 1;
   87 
   88 @iva = ($iv0, $iv1, $iv2);
   89 @jva = ($jv0, $jv1, $jv2);
   90 @kva = ($kv0, $kv1, $kv2);
   91 
   92 $iv = Vector(@iva);
   93 $jv = Vector(@jva);
   94 $kv = Vector(@kva);
   95 
   96 $v0 = Real(random(-10, 10, 1));
   97 $v1 = Real(random(-10, 10, 1));
   98 $v2 = Real(random(-10, 10, 1));
   99 
  100 # make unit vector
  101 #$v0 = $v0 / vlen($v0, $v1, $v2);
  102 #$v1 = $v1 / vlen($v0, $v1, $v2);
  103 #$v2 = $v2 / vlen($v0, $v1, $v2);
  104 
  105 
  106 $w0 = Real(random(-10, 10, 1));
  107 $w1 = Real(random(-10, 10, 1));
  108 $w2 = Real(random(-10, 10, 1));
  109 
  110 # make unit vector
  111 #$w0 = $v0 / vlen($v0, $v1, $v2);
  112 #$w1 = $v1 / vlen($v0, $v1, $v2);
  113 #$w2 = $v2 / vlen($v0, $v1, $v2);
  114 
  115 @va = ($v0, $v1, $v2);
  116 @wa = ($w0, $w1, $w2);
  117 
  118 $v = Vector($v0, $v1, $v2);
  119 $w = Vector($w0, $w1, $w2);
  120 
  121 $ans = vdot(@va, @wa);
  122 
  123 Context()->texStrings;
  124 BEGIN_TEXT
  125 \{ beginproblem() \}
  126 \{ textbook_ref_exact("Rogawski ET 2e", "12.3","11") \}
  127 $PAR
  128 \( v = $v \)
  129 $PAR
  130 \( w = $w \)
  131 $PAR
  132 Compute the dot product \( v \cdotp w \).
  133 $PAR
  134 \{ans_rule()\}
  135 END_TEXT
  136 Context()->normalStrings;
  137 
  138 ANS($ans->cmp);
  139 
  140 Context()->texStrings;
  141 SOLUTION(EV3(<<'END_SOLUTION'));
  142 $PAR
  143 $SOL
  144 \( v \cdotp w = ($v) \cdotp ($w) \) $BR
  145 \(\quad = ($v0 \cdot $w0) + ($v1 \cdot $w1) + ($v2 \cdot $w2) = $ans \)
  146 END_SOLUTION
  147 
  148 ENDDOCUMENT();
  149 
  150