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Tue Nov 8 15:17:41 2011 UTC (18 months, 1 week ago) by aubreyja
File size: 4336 byte(s)
Rogawski problems contributed by publisher WHFreeman. These are a subset of the problems available to instructors who use the Rogawski textbook. The remainder can be obtained from the publisher.

    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('46')
10 ## Author('Christopher Sira')
11 ## Institution('W.H.Freeman')
12
13 DOCUMENT();
18 $context = Context("Vector"); 19 20 sub vdot 21 { 22 my($temp) = 0;
23
24     for ($i = 0;$i < 3; $i++) 25 { 26$temp = $temp +$_[$i] *$_[$i + 3]; 27 } 28 return$temp
29 }
30
31 sub vlen
32 {
33     return sqrt($_[0]**2 +$_[1]**2 + $_[2]**2); 34 } 35 36 sub vcos 37 { 38 my($dot) = vdot(@_);
39
40     return $dot / (vlen($_[0], $_[1],$_[2]) * vlen($_[3],$_[4], $_[5])); 41 } 42 43 sub vang 44 { 45 return Formula("arccos(x)")->eval(x=>vcos(@_)); 46 } 47 48 # projection of u onto v (first onto second) 49 sub vproj 50 { 51 my(@temp) = ($_[3], $_[4],$_[5]);
52     my($temp2) = vdot(@_); 53 54$temp2 = $temp2 / vdot($_[3], $_[4],$_[5], $_[3],$_[4], $_[5]); 55 56$temp[0] *= $temp2; 57$temp[1] *= $temp2; 58$temp[2] *= $temp2; 59 60 return @temp; 61 } 62 63 # perpendicular component of u onto v (first onto second) 64 sub vperp 65 { 66 my(@temp) = vproj(@_); 67 68$temp[0] = -1 * $temp[0] +$_[0];
69     $temp[1] = -1 *$temp[1] + $_[1]; 70$temp[2] = -1 * $temp[2] +$_[2];
71
72     return @temp;
73 }
74
75 $rad = Real(random(0.1, pi, 0.1)); 76$lenv = Real(random(1, 5, 1));
77 $lenw = Real(random(1, 5, 1)); 78 79$a1 = Real(random(1, 4, 1));
80 $b1 = Real(random(1, 4, 1)); 81 82$a2 = Real(random(1, 4, 1));
83 $b2 = Real(random(-4, -1, 1)); 84 85 86$ans1 = Formula("$lenv *$lenw * cos($rad)")->eval(); 87$ans2 = sqrt($a1**2 *$lenv**2 + 2 * $a1 *$b1 * $ans1 +$b1**2 * $lenw**2); 88$ans3 = sqrt($a2**2 *$lenv**2 + 2 * $a2 *$b2 * $ans1 +$b2**2 * $lenw**2); 89 90 Context()->texStrings; 91 BEGIN_TEXT 92 \{ beginproblem() \} 93 \{ textbook_ref_exact("Rogawski ET 2e", "12.3","46") \} 94$PAR
95 $$\| v \| = lenv$$
96 $PAR 97 $$\| w \| = lenw$$ 98$PAR
99 The angle between $$v$$ and $$w$$ is $rad radians. 100$PAR
101 Given this information, calculate the following:
102 $PAR 103 (a) $$v \cdotp w$$ = \{ans_rule()\} 104$PAR
105 (b) $$\| a1 v + b1 w \| =$$ \{ans_rule()\}
106 $PAR 107 (c) $$\| a2 v + b2 w \| =$$ \{ans_rule()\} 108$PAR
109 END_TEXT
110 Context()->normalStrings;
111
112 ANS($ans1->cmp,$ans2->cmp, $ans3->cmp); 113 114 Context()->texStrings; 115 SOLUTION(EV3(<<'END_SOLUTION')); 116$PAR
117 $SOL 118$PAR
119 (a) We know that $$v \cdotp w = \| v \| \| w \| \cos{\theta}$$, so
120 $PAR 121 $$v \cdotp w = lenv (lenw) \cos{(rad)} = ans1$$. 122$PAR
123 (b) Since $$u \cdotp u = {\| u \|}^{2}$$,
124 $PAR 125 we know that $$\| a1 v + b1 w \| = \sqrt{(a1 v + b1 w) \cdotp (a1 v + b1 w)}$$. 126$PAR
127 Multiplying this out (using the distributive property of dot products) we get:
128 $PAR 129 $$\| a1 v + b1 w \| = \sqrt{ \{a1**2\}(v \cdotp v) + \{2 * a1 * b1\}(v \cdotp w) + \{b1**2\}(w \cdotp w)}$$. 130$PAR
131 We can simplify this further because we know $$\| w \| = lenw$$ and $$\| v \| = lenv$$ and $$v \cdotp w = ans1$$ (from part a):
132 $PAR 133 $$\| a1 v + b1 w \| = \sqrt{ \{a1**2\}({lenv}^2) + \{2 * a1 * b1\}(ans1) + \{b1**2\}({lenw}^2)}$$. 134$PAR
135 And finally,
136 $PAR 137 $$\| a1 v + b1 w \| = \sqrt{ \{a1**2 * lenv**2 + 2 * a1 * b1 * ans1 + b1**2 * lenw**2\} } = ans2$$. 138$PAR
139 (c) Since $$u \cdotp u = {\| u \|}^{2}$$,
140 $PAR 141 we know that $$\| a2 v + b2 w \| = \sqrt{(a2 v + b2 w) \cdotp (a2 v + b2 w)}$$. 142$PAR
143 Multiplying this out (using the distributive property of dot products) we get:
144 $PAR 145 $$\| a2 v + b2 w \| = \sqrt{ \{a2**2\}(v \cdotp v) + \{2 * a2 * b2\}(v \cdotp w) + \{b2**2\}(w \cdotp w)}$$. 146$PAR
147 We can simplify this further because we know $$\| w \| = lenw$$ and $$\| v \| = lenv$$ and $$v \cdotp w = ans1$$ (from part a):
148 $PAR 149 $$\| a2 v + b2 w \| = \sqrt{ \{a2**2\}({lenv}^2) + \{2 * a2 * b2\}(ans1) + \{b2**2\}({lenw}^2)}$$. 150$PAR
151 And finally,
152 \$PAR
153 $$\| a2 v + b2 w \| = \sqrt{ \{a2**2 * lenv**2 + 2 * a2 * b2 * ans1 + b2**2 * lenw**2\} } = ans3$$.
154 END_SOLUTION
155
156 ENDDOCUMENT();
157
158