View of /trunk/NationalProblemLibrary/TCNJ/TCNJ_LengthOrthogonality/problem6.pg

    1 ## DESCRIPTION
    2 ## Linear Algebra
    3 ## ENDDESCRIPTION
    4 
    5 ## KEYWORDS('inner product' 'orthogonal' 'angle')
    6 ## Tagged by tda2d
    7 
    8 ## DBsubject('Linear Algebra')
    9 ## DBchapter('Vector Spaces')
   10 ## DBsection('Inner Product')
   11 ## Date('')
   12 ## Author('')
   13 ## Institution('TCNJ')
   14 ## TitleText1('')
   15 ## EditionText1('')
   16 ## AuthorText1('')
   17 ## Section1('')
   18 ## Problem1('')
   19 
   20 DOCUMENT();        # This should be the first executable line in the problem.
   21 
   22 loadMacros(
   23 "PG.pl",
   24 "PGbasicmacros.pl",
   25 "PGanswermacros.pl",
   26 "PGauxiliaryFunctions.pl",
   27 "PGmatrixmacros.pl"
   28 );
   29 
   30 TEXT(beginproblem());
   31 $showPartialCorrectAnswers = 1;
   32 
   33 @x=(0,0);
   34 @y=(0,0);
   35 
   36 while($x[0]*$y[1]==$x[1]*$y[0])
   37 {
   38 $prod = 0;
   39 $norm_x = 0;
   40 $norm_y = 0;
   41 foreach $i (0..1) {
   42   $x[$i] = non_zero_random(-5,5,1);
   43   $y[$i] = non_zero_random(-5,5,1);
   44   if ($y[$i] == $x[$i]) {
   45     $y[$i] = 6;
   46   }
   47   $prod += $x[$i] * $y[$i];
   48     $norm_x += $x[$i] * $x[$i];
   49     $norm_y += $y[$i] * $y[$i];
   50 }
   51 }
   52 
   53 $norm_x = sqrt($norm_x);
   54 $norm_y = sqrt($norm_y);
   55 
   56 $angle = arccos( $prod / $norm_x / $norm_y);
   57 
   58 BEGIN_TEXT
   59 Find the angle \( \alpha \) between the vectors
   60 \{ mbox( display_matrix([[$x[0]], [$x[1]]]), ' and ', display_matrix([[$y[0]], [$y[1]]]), '.' ) \}
   61 $BR
   62 \( \alpha = \) \{ans_rule(20)\}.
   63 
   64 END_TEXT
   65 
   66 ANS(num_cmp($angle));
   67 
   68 
   69 ENDDOCUMENT();       # This should be the last executable line in the problem.
   70