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| 1 : | ted shifri | 1457 | ## DESCRIPTION |
| 2 : | ## Use Cross Products to Find Equation of a Plane through Three Points | ||
| 3 : | ## ENDDESCRIPTION | ||
| 4 : | |||
| 5 : | ## KEYWORDS('Cross Product', 'Equation of Plane') | ||
| 6 : | ## | ||
| 7 : | |||
| 8 : | ## DBsubject('Calculus') | ||
| 9 : | ## DBchapter('Vectors and the Geometry of Space') | ||
| 10 : | ## DBsection('The Cross Product') | ||
| 11 : | ## Date('8/22/2009') | ||
| 12 : | ## Author('Ted Shifrin') | ||
| 13 : | ## Institution('UGA') | ||
| 14 : | ## TitleText1('') | ||
| 15 : | ## EditionText1('') | ||
| 16 : | ## AuthorText1('') | ||
| 17 : | ## Section1('') | ||
| 18 : | ## Problem1('') | ||
| 19 : | |||
| 20 : | DOCUMENT(); # This should be the first executable line in the problem. | ||
| 21 : | |||
| 22 : | loadMacros("PG.pl", | ||
| 23 : | "PGbasicmacros.pl", | ||
| 24 : | "PGchoicemacros.pl", | ||
| 25 : | "PGanswermacros.pl", | ||
| 26 : | "PGauxiliaryFunctions.pl", | ||
| 27 : | "PGmatrixmacros.pl" | ||
| 28 : | ); | ||
| 29 : | |||
| 30 : | TEXT(beginproblem()); | ||
| 31 : | $showPartialCorrectAnswers = 1; | ||
| 32 : | |||
| 33 : | $a = random(-5, 5, 1); | ||
| 34 : | $b = random(-5, 5, 1); | ||
| 35 : | $c = random(-5, 5, 1); | ||
| 36 : | $d = random(-2, 2, 1); | ||
| 37 : | $e = random(-3, 3, 1); | ||
| 38 : | $f = random(-3,3,1); | ||
| 39 : | $g = $a+$d; | ||
| 40 : | $h = $b+$e; | ||
| 41 : | $i = $c+1; | ||
| 42 : | $j = $g+$f; | ||
| 43 : | $k = $h+$e*$f-1; | ||
| 44 : | $l = $i+$f; | ||
| 45 : | |||
| 46 : | $ans1 = (1-$d)*$f; | ||
| 47 : | $ans2 = ($d-1)*($e*$f-1)-1; | ||
| 48 : | $ans3 = $a+$ans1*$b+$ans2*$c; | ||
| 49 : | |||
| 50 : | |||
| 51 : | BEGIN_TEXT | ||
| 52 : | Find a vector with first coordinate equal to \( 1\) that is orthogonal to | ||
| 53 : | the plane through the points | ||
| 54 : | \{ mbox( display_matrix([[$a],[$b],[$c]]),",", display_matrix([[$g],[$h],[$i]]) , ",", "\(\quad\) and", display_matrix([[$j],[$k],[$l]]), ".") \} | ||
| 55 : | $BR | ||
| 56 : | \{ mbox( answer_matrix(3,1,6) ) \} | ||
| 57 : | $BR | ||
| 58 : | |||
| 59 : | |||
| 60 : | Give an equation of the plane passing through the same three points: | ||
| 61 : | $BR | ||
| 62 : | \( x \)+ \{ans_rule(6)\}\( y + \) \{ans_rule(6)\}\( z= \) \{ans_rule(6)\}. $BR | ||
| 63 : | (Make sure to fill in each box, even if a coefficient might happen to be \(0\).) $BR | ||
| 64 : | END_TEXT | ||
| 65 : | |||
| 66 : | ANS(num_cmp(1)); | ||
| 67 : | ANS(num_cmp($ans1)); | ||
| 68 : | ANS(num_cmp($ans2)); | ||
| 69 : | |||
| 70 : | ANS(num_cmp($ans1)); | ||
| 71 : | ANS(num_cmp($ans2)); | ||
| 72 : | ANS(num_cmp($ans3)); | ||
| 73 : | |||
| 74 : | ENDDOCUMENT(); # This should be the last executable line in the problem. |
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