## DESCRIPTION
##   Dot Product 
## ENDDESCRIPTION

## KEYWORDS('Dot Product', 'Angle')


## DBsubject('Calculus')
## DBchapter('Vectors and the Geometry of Space')
## DBsection('Dot Product')
## Date('5/27/10')
## Author('Ted Shifrin'
## Institution('UGA')
## TitleText1('')
## EditionText1('')
## AuthorText1('')
## Section1('')
## Problem1('')
           
DOCUMENT();	

loadMacros("PG.pl",
           "PGbasicmacros.pl",
           "PGchoicemacros.pl",
           "PGanswermacros.pl",
           "PGauxiliaryFunctions.pl",
           "PGmatrixmacros.pl",
           "PGmorematrixmacros.pl",
           "Parser.pl",
           "MathObjects.pl",
           );

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

Context("Vector");


$a = random(1,4);
do{$b = random(1,4)} until ($b != $a);

do{$c = non_zero_random(-3,3)} until ($c**2!=1);
do{$d = non_zero_random(-3,3)} until ($d!=$c && $d!=-$c && $d**2!=1);

$m = $a*$b*($c+$d);

$ans1 = Real("sqrt(($a)^2+($c*$b)^2+ 2*$c/($c+$d))");
$ans2 = Real("sqrt(($a)^2+($b*$d)^2+ 2*$d/($c+$d))");
$ans3 = Real("arccos((($a)^2+$c*$d*($b)^2 + 1)/($ans1*$ans2))");

Context()->texStrings;
BEGIN_TEXT

If \(\|\mathbf x\| = $a\), \(\|\mathbf y\| = $b\), and the angle between \(\mathbf x\) and \(\mathbf y\) is \(\theta = \arccos(1/$m)\), then
$PAR
(a) \(\|\mathbf x + $c\mathbf y\| = \) \{ans_rule(15)\}
$PAR
(b) \(\|\mathbf x + $d\mathbf y\| = \) \{ans_rule(15)\}
$PAR
(c) and the angle
between the vectors \(\mathbf x + $c\mathbf y\) and \(\mathbf x + $d\mathbf y\) is (in radians) \{ans_rule(20)\}


END_TEXT

ANS($ans1->cmp,$ans2->cmp,$ans3->cmp);

ENDDOCUMENT();