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| 1 : | aubreyja | 2584 | ## DBsubject('Calculus') |
| 2 : | ## DBchapter('Introduction to Differential Equations') | ||
| 3 : | ## DBsection('Vectors in Three Dimensions') | ||
| 4 : | ## KEYWORDS('calculus', 'parametric', 'vector', '3D', 'three dimensions') | ||
| 5 : | ## TitleText1('Calculus: Early Transcendentals') | ||
| 6 : | ## EditionText1('2') | ||
| 7 : | ## AuthorText1('Rogawski') | ||
| 8 : | ## Section1('12.2') | ||
| 9 : | ## Problem1('27') | ||
| 10 : | ## Author('Christopher Sira') | ||
| 11 : | ## Institution('W.H.Freeman') | ||
| 12 : | |||
| 13 : | DOCUMENT(); | ||
| 14 : | loadMacros("PG.pl","PGbasicmacros.pl","PGanswermacros.pl"); | ||
| 15 : | loadMacros("PGchoicemacros.pl"); | ||
| 16 : | loadMacros("Parser.pl"); | ||
| 17 : | loadMacros("freemanMacros.pl"); | ||
| 18 : | $context = Context("Vector"); | ||
| 19 : | |||
| 20 : | $a = Real(random(1,10,1)); | ||
| 21 : | $b = Real(random(1,10,1)); | ||
| 22 : | $c = Real(random(1,10,1)); | ||
| 23 : | |||
| 24 : | $v = Vector($a, $b, $c); | ||
| 25 : | $w = -1 * $v; | ||
| 26 : | |||
| 27 : | $len = sqrt($a**2 + $b**2 + $c**2); | ||
| 28 : | |||
| 29 : | $ans = Vector($a / $len, $b / $len, $c / $len); | ||
| 30 : | |||
| 31 : | |||
| 32 : | Context()->texStrings; | ||
| 33 : | BEGIN_TEXT | ||
| 34 : | \{ beginproblem() \} | ||
| 35 : | \{ textbook_ref_exact("Rogawski ET 2e", "12.2","27") \} | ||
| 36 : | $PAR | ||
| 37 : | Find a unit vector \( \mathbf{u} \) in the direction opposite of \( $w \). | ||
| 38 : | $PAR | ||
| 39 : | \( \mathbf{u} \) = \{ans_rule()\} | ||
| 40 : | END_TEXT | ||
| 41 : | Context()->normalStrings; | ||
| 42 : | |||
| 43 : | ANS($ans->cmp); | ||
| 44 : | |||
| 45 : | Context()->texStrings; | ||
| 46 : | SOLUTION(EV3(<<'END_SOLUTION')); | ||
| 47 : | $PAR | ||
| 48 : | $SOL | ||
| 49 : | \( \mathbf{w} = $w \) | ||
| 50 : | $PAR | ||
| 51 : | \( \mathbf{v} = -1 * $w = $v \) | ||
| 52 : | $PAR | ||
| 53 : | \( \| \mathbf{v} \| = \sqrt{\mathbf{v} \cdotp \mathbf{v}} = \sqrt{$a * $a + $b * $b + $c * $c} = $len \). | ||
| 54 : | $PAR | ||
| 55 : | \( \mathbf{u} = \mathbf{e}_\mathbf{v} = \frac{1}{ \| \mathbf{v} \|} v = \left< \frac{$a}{$len}, | ||
| 56 : | \frac{$b}{$len}, \frac{$c}{$len} \right> = $ans \) | ||
| 57 : | END_SOLUTION | ||
| 58 : | |||
| 59 : | ENDDOCUMENT(); | ||
| 60 : | |||
| 61 : |
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