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# Annotation of /branches/UGA/1.2.4.pg

 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();