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drdrew42drdrew42
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Merge pull request #1250 from brunners23/main
feat: Add Georgia Tech Problems
2 parents c1d4874 + 593af81 commit 983a829

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‎Contrib/GeorgiaTech/LengthsAndDotProducts.pg‎

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##DESCRIPTION
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## Orthogonal sets, matrices, and projections
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##ENDDESCRIPTION
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## DBsubject(Linear algebra)
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## DBchapter(Vector geometry)
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## DBsection(Dot product, length, and unit vectors)
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## Date(11/25/2024)
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## Institution(Georgia Institute of Technology)
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## Author(Seth Brunner)
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## Level(2)
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DOCUMENT();
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loadMacros(
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"PGstandard.pl",
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"PGgraders.pl",
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"MathObjects.pl",
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"PGchoicemacros.pl",
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"matrixExtensions.pl",
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"unionLists.pl",
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"unionTables.pl",
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"Alfredmacros.pl",
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"parserPopUp.pl",
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"PGcourse.pl",
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);
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TEXT(beginproblem());
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Context("Numeric");
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$dropdown_a = PopUp(["?", "A", "B", "C", "D","More than one of the given sets","None of the given sets"], "B");
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$dropdown_b = PopUp(["?", "A", "B", "C", "D","More than one of the given sets","None of the given sets"], "D");
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$dropdown_c = PopUp(["?", "A", "B", "C", "D","More than one of the given sets","None of the given sets"], "C");
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$dropdown_d = PopUp(["?", "A", "B", "C", "D","More than one of the given sets","None of the given sets"], "A");
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$normu=random(1,5,1);
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$udotu=Compute("$normu**2");
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$normv=random(1,6,1);
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$vdotv=Compute("$normv**2");
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$ssl=Compute("$normu**2+$normv**2");
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while($normu==$normv)
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{
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$normv=random(1,6,1);
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}
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$dotP=0;
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Context()->texStrings;
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$mc1 = new_checkbox_multiple_choice();
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$mc1 -> qa (
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"Which of the following sets contain only unit vectors? $BITALIC Select all that apply$EITALIC.",
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"\(\left\lbrace\frac1{$normu}\vec u,\frac1{$normv}\vec v\right\rbrace\)",
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"\(\left\lbrace\frac{$normv}{$udotu\sqrt{$ssl}}\vec u+\frac{$normu}{$vdotv\sqrt{$ssl}}\vec v,\frac{$normu}{$udotu\sqrt{$ssl}}\vec u+\frac{$normv}{$vdotv\sqrt{$ssl}}\vec v\right\rbrace\)",
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);
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$mc1 -> extra(
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"\(\left\lbrace\vec u+\vec v,\vec u-\vec v\right\rbrace\)",
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"\(\left\lbrace\frac1{$normu}\vec u+\frac1{$normv}\vec v,\frac1{$normu}\vec u-\frac1{$normv}\vec v\right\rbrace\)",
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);
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$mc1 -> makeLast("None of the above");
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$mc2 = new_checkbox_multiple_choice();
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$mc2 -> qa (
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"Which of the following sets are orthogonal? $BITALIC Select all that apply$EITALIC.",
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"\(\left\lbrace\frac1{$normu}\vec u,\frac1{$normv}\vec v\right\rbrace\)",
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"\(\left\lbrace\frac1{$normu}\vec u+\frac1{$normv}\vec v,\frac1{$normu}\vec u-\frac1{$normv}\vec v\right\rbrace\)",
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);
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$mc2 -> extra(
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"\(\left\lbrace\vec u+\vec v,\vec u-\vec v\right\rbrace\)",
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"\(\left\lbrace\frac{$normv}{$udotu\sqrt{$ssl}}\vec u+\frac{$normu}{$vdotv\sqrt{$ssl}}\vec v,\frac{$normu}{$udotu\sqrt{$ssl}}\vec u+\frac{$normv}{$vdotv\sqrt{$ssl}}\vec v\right\rbrace\)",
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);
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$mc2 -> makeLast("None of the above");
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$mc3 = new_checkbox_multiple_choice();
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$mc3 -> qa (
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"Based on the previous two questions, which of the following sets are orthonormal? $BITALIC Select all that apply$EITALIC.",
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"\(\left\lbrace\frac1{$normu}\vec u,\frac1{$normv}\vec v\right\rbrace\)",
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);
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$mc3 -> extra(
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"\(\left\lbrace\vec u+\vec v,\vec u-\vec v\right\rbrace\)",
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"\(\left\lbrace\frac{$normv}{$udotu\sqrt{$ssl}}\vec u+\frac{$normu}{$vdotv\sqrt{$ssl}}\vec v,\frac{$normu}{$udotu\sqrt{$ssl}}\vec u+\frac{$normv}{$vdotv\sqrt{$ssl}}\vec v\right\rbrace\)",
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"\(\left\lbrace\frac1{$normu}\vec u+\frac1{$normv}\vec v,\frac1{$normu}\vec u-\frac1{$normv}\vec v\right\rbrace\)",
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);
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$mc3 -> makeLast("None of the above");
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BEGIN_TEXT
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<p>
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Suppose \(\vec u\) and \(\vec v\) are vectors with \(\left\|\vec u\right\|=$normu\), \(\left\|\vec v\right\|=$normv\), and \(\vec u\cdot\vec v=$dotP\).
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</p>
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$BR
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\{ $mc1 -> print_q() \}
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$BR
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\{ $mc1 -> print_a() \}
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$BR
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$BR
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\{ $mc2 -> print_q() \}
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$BR
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\{ $mc2 -> print_a() \}
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$BR
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$BR
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\{ $mc3 -> print_q() \}
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$BR
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\{ $mc3 -> print_a() \}
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END_TEXT
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Context()->normalStrings;
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##################################################
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# Answers
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$showPartialCorrectAnswers = 0;
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install_problem_grader(~~&std_problem_grader);
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$showPartialCorrectAnswers = 0;
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ANS(checkbox_cmp($mc1->correct_ans()));
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ANS(checkbox_cmp($mc2->correct_ans()));
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ANS(checkbox_cmp($mc3->correct_ans()));
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ENDDOCUMENT();

‎Contrib/GeorgiaTech/Module1TrueorFalse.pg‎

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## DBsubject(Calculus - multivariable)
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## DBchapter(Calculus of vector valued functions)
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## DBsection(Parameterized curves)
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## Institution(Georgia Institute of Technology)
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## Author(Gregory Mayer, Hunter Lehman)
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## Level(2)
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## MO(1)
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## Language(en)
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DOCUMENT();
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loadMacros(
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"PGstandard.pl",
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"MathObjects.pl",
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"PGchoicemacros.pl",
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"unionLists.pl",
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"parserPopUp.pl",
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"PGcourse.pl",
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);
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TEXT(beginproblem(), $BR,$BBOLD, "True or False Exercise", $EBOLD, $BR,$BR);
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$showPartialCorrectAnswers = 0;
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@questions = (
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"The speed of a particle with a position function \(r(t)\) is \(\frac{r'(t)}{|r'(t)|}\).",
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"If \(u\) and \(v\) are two vectors in \(\mathbb{R}^3\), then \(u \cdot v\) is a vector orthogonal to both \(u\) and \(v\).",
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"The point (2, -1, 3) lies on the graph of the sphere \((x - 2)^2 + (y + 1)^2 + (z - 3)^2 = 25\).",
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"\(\frac{d}{dt}[u(t) \times u(t)] = 2u'(t) \times u(t)\). ",
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"Parallel planes have parallel normal vectors. ",
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"The vector \(\frac{1}{\sqrt{3}} \hat i - \frac{1}{\sqrt{3}} \hat j + \frac{2}{\sqrt{3}} \hat k\) is a unit vector. ",
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"The only way that \(v \cdot w\) = 0 is if \(v\) = 0 or \(w\) = 0.",
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);
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@corans = ('False', 'False', 'False', 'False', 'True', 'True', 'False');
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# Choose Seven of the question and answer pairs at random.
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@which = NchooseK(7, 7);
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@squestions = @questions[@which];
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@scorans = @corans[@which];
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foreach my $i (0..6) {
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$popup[$i] = PopUp(['Choose','True', 'False'],$scorans[$i]);
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}
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BEGIN_TEXT
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Indicate whether the following statements are true or false.
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$PAR
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\{ BeginList('OL',type=>'a') \}
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$ITEM
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$squestions[0]
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$BR
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\{ $popup[0]->menu \}
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$ITEMSEP
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$ITEM
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$squestions[1]
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$BR
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\{ $popup[1]->menu \}
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$ITEMSEP
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$ITEM
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$squestions[2]
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$BR
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\{ $popup[2]->menu \}
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$ITEMSEP
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$ITEM
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$squestions[3]
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$BR
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\{ $popup[3]->menu \}
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$ITEMSEP
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$ITEM
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$squestions[4]
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$BR
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\{ $popup[4]->menu \}
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$ITEMSEP
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$ITEM
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$squestions[5]
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$BR
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\{ $popup[5]->menu \}
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$ITEMSEP
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$ITEM
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$squestions[6]
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$BR
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\{ $popup[6]->menu \}
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$ITEMSEP
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\{ EndList('OL') \}
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END_TEXT
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install_problem_grader(~~&std_problem_grader);
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# Enter the correct answers to be checked against the answers to the students.
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foreach my $i (0..6) {
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ANS($popup[$i]->cmp);
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}
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Context()->texStrings;
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@solutions = (
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"$BBOLD False $EBOLD
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$BR
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The speed of a particle is defined as the magnitude of its velocity vector, not the normalized velocity. The correct formula for speed is \(|r'(t)|\), where \(r'(t)\) is the velocity vector. The given expression \(\frac{r'(t)}{|r'(t)|}\) would always have a magnitude of 1, representing only the direction of motion, not the speed. ",
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"$BBOLD False $EBOLD
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$BR
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\(u \cdot v\) is a scalar (dot product), not a vector. The cross product \(u × v\) would give a vector orthogonal to both u and v, but not the dot product.",
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"$BBOLD False $EBOLD
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$BR
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The sphere equation represents points whose distance from the center (2, -1, 3) equals the radius \(\sqrt{25} = 5\). Substituting (2, -1, 3) into the equation results in all squared terms becoming zero. Since any distance squared is non-negative, the equation reduces to \(0 = 25\), which is never true. This mathematical contradiction confirms (2, -1, 3) is not on the sphere. ",
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"$BBOLD False $EBOLD
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$BR
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The correct differentiation of \(u(t) \times u(t)\) (cross product of a vector with itself) with respect to \(t\) is \(u'(t) \times u(t) + u(t) \times u'(t) \). Note also that the cross product of a vector with itself happens to be zero. But either way the statement is false because it does not use the correct differentiation formula.",
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"$BBOLD True $EBOLD
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$BR
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Parallel planes have identical normal vectors or normal vectors that are scalar multiples of each other. ",
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"$BBOLD True $EBOLD
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$BR
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Magnitude = \(\sqrt{\left(\frac{1}{\sqrt{3}}\right)^2 + \left(\frac{1}{\sqrt{3}}\right)^2 + \left(\frac{2}{\sqrt{3}}\right)^2} = 1\). A vector with magnitude 1 is a unit vector. ",
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"$BBOLD False $EBOLD
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$BR
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Two non-zero orthogonal vectors can have a dot product of zero.",
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);
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$solutions = "";
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foreach (@which) {
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$solutions .= "$ITEM $solutions[$_]";
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}
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SOLUTION(EV3(<<'END_SOLUTION'));
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\{ BeginList('OL',type=>'a') \}
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$solutions
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\{ EndList('OL') \}
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END_SOLUTION
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Context()->normalStrings;
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ENDDOCUMENT(); # This should be the last executable line in the problem.

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