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| 1 : | ted shifri | 1457 | ## DESCRIPTION |
| 2 : | ## Dot Product | ||
| 3 : | ## ENDDESCRIPTION | ||
| 4 : | |||
| 5 : | ## KEYWORDS('Dot Product', 'Angle') | ||
| 6 : | |||
| 7 : | |||
| 8 : | ## DBsubject('Calculus') | ||
| 9 : | ## DBchapter('Vectors and the Geometry of Space') | ||
| 10 : | ## DBsection('Dot Product') | ||
| 11 : | ## Date('5/27/10') | ||
| 12 : | ## Author('Ted Shifrin' | ||
| 13 : | ## Institution('UGA') | ||
| 14 : | ## TitleText1('') | ||
| 15 : | ## EditionText1('') | ||
| 16 : | ## AuthorText1('') | ||
| 17 : | ## Section1('') | ||
| 18 : | ## Problem1('') | ||
| 19 : | |||
| 20 : | DOCUMENT(); | ||
| 21 : | |||
| 22 : | loadMacros("PG.pl", | ||
| 23 : | "PGbasicmacros.pl", | ||
| 24 : | "PGchoicemacros.pl", | ||
| 25 : | "PGanswermacros.pl", | ||
| 26 : | "PGauxiliaryFunctions.pl", | ||
| 27 : | "PGmatrixmacros.pl", | ||
| 28 : | "PGmorematrixmacros.pl", | ||
| 29 : | "Parser.pl", | ||
| 30 : | "MathObjects.pl", | ||
| 31 : | ); | ||
| 32 : | |||
| 33 : | TEXT(beginproblem()); | ||
| 34 : | $showPartialCorrectAnswers = 1; | ||
| 35 : | |||
| 36 : | Context("Vector"); | ||
| 37 : | |||
| 38 : | |||
| 39 : | $a = random(1,4); | ||
| 40 : | do{$b = random(1,4)} until ($b != $a); | ||
| 41 : | |||
| 42 : | do{$c = non_zero_random(-3,3)} until ($c**2!=1); | ||
| 43 : | do{$d = non_zero_random(-3,3)} until ($d!=$c && $d!=-$c && $d**2!=1); | ||
| 44 : | |||
| 45 : | $m = $a*$b*($c+$d); | ||
| 46 : | |||
| 47 : | $ans1 = Real("sqrt(($a)^2+($c*$b)^2+ 2*$c/($c+$d))"); | ||
| 48 : | $ans2 = Real("sqrt(($a)^2+($b*$d)^2+ 2*$d/($c+$d))"); | ||
| 49 : | $ans3 = Real("arccos((($a)^2+$c*$d*($b)^2 + 1)/($ans1*$ans2))"); | ||
| 50 : | |||
| 51 : | Context()->texStrings; | ||
| 52 : | BEGIN_TEXT | ||
| 53 : | |||
| 54 : | If \(\|\mathbf x\| = $a\), \(\|\mathbf y\| = $b\), and the angle between \(\mathbf x\) and \(\mathbf y\) is \(\theta = \arccos(1/$m)\), then | ||
| 55 : | $PAR | ||
| 56 : | (a) \(\|\mathbf x + $c\mathbf y\| = \) \{ans_rule(15)\} | ||
| 57 : | $PAR | ||
| 58 : | (b) \(\|\mathbf x + $d\mathbf y\| = \) \{ans_rule(15)\} | ||
| 59 : | $PAR | ||
| 60 : | (c) and the angle | ||
| 61 : | between the vectors \(\mathbf x + $c\mathbf y\) and \(\mathbf x + $d\mathbf y\) is (in radians) \{ans_rule(20)\} | ||
| 62 : | |||
| 63 : | |||
| 64 : | END_TEXT | ||
| 65 : | |||
| 66 : | ANS($ans1->cmp,$ans2->cmp,$ans3->cmp); | ||
| 67 : | |||
| 68 : | ENDDOCUMENT(); |
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