feat: Add Georgia Tech Problems

Contrib/GeorgiaTech/LengthsAndDotProducts.pg

@@ -0,0 +1,148 @@
+##DESCRIPTION
+##  Orthogonal sets, matrices, and projections
+##ENDDESCRIPTION
+
+
+## DBsubject(Linear algebra)
+## DBchapter(Vector geometry)
+## DBsection(Dot product, length, and unit vectors)
+## Date(11/25/2024)
+## Institution(Georgia Institute of Technology)
+## Author(Seth Brunner)
+## Level(2)
+
+DOCUMENT();      
+loadMacros(
+    "PGstandard.pl",
+    "PGgraders.pl",
+    "MathObjects.pl",
+    "PGchoicemacros.pl",
+    "matrixExtensions.pl",
+    "unionLists.pl",
+    "unionTables.pl",
+    "Alfredmacros.pl",
+    "parserPopUp.pl",
+    "PGcourse.pl",
+);
+
+
+TEXT(beginproblem());
+
+Context("Numeric");
+
+$dropdown_a = PopUp(["?", "A", "B", "C", "D","More than one of the given sets","None of the given sets"], "B");
+$dropdown_b = PopUp(["?", "A", "B", "C", "D","More than one of the given sets","None of the given sets"], "D");
+$dropdown_c = PopUp(["?", "A", "B", "C", "D","More than one of the given sets","None of the given sets"], "C");
+$dropdown_d = PopUp(["?", "A", "B", "C", "D","More than one of the given sets","None of the given sets"], "A");
+
+$normu=random(1,5,1);
+$udotu=Compute("$normu**2");
+$normv=random(1,6,1);
+$vdotv=Compute("$normv**2");
+$ssl=Compute("$normu**2+$normv**2");
+while($normu==$normv)
+{
+    $normv=random(1,6,1);
+}
+$dotP=0;
+
+Context()->texStrings;
+
+$mc1 = new_checkbox_multiple_choice();
+$mc1 -> qa (
+"Which of the following sets contain only unit vectors? $BITALIC Select all that apply$EITALIC.", 
+
+"\(\left\lbrace\frac1{$normu}\vec u,\frac1{$normv}\vec v\right\rbrace\)",
+
+"\(\left\lbrace\frac{$normv}{$udotu\sqrt{$ssl}}\vec u+\frac{$normu}{$vdotv\sqrt{$ssl}}\vec v,\frac{$normu}{$udotu\sqrt{$ssl}}\vec u+\frac{$normv}{$vdotv\sqrt{$ssl}}\vec v\right\rbrace\)",
+    );
+$mc1 -> extra(
+"\(\left\lbrace\vec u+\vec v,\vec u-\vec v\right\rbrace\)",
+
+"\(\left\lbrace\frac1{$normu}\vec u+\frac1{$normv}\vec v,\frac1{$normu}\vec u-\frac1{$normv}\vec v\right\rbrace\)",
+);
+$mc1 -> makeLast("None of the above");
+
+$mc2 = new_checkbox_multiple_choice();
+$mc2 -> qa (
+"Which of the following sets are orthogonal? $BITALIC Select all that apply$EITALIC.", 
+
+"\(\left\lbrace\frac1{$normu}\vec u,\frac1{$normv}\vec v\right\rbrace\)",
+
+"\(\left\lbrace\frac1{$normu}\vec u+\frac1{$normv}\vec v,\frac1{$normu}\vec u-\frac1{$normv}\vec v\right\rbrace\)",
+    );
+$mc2 -> extra(
+"\(\left\lbrace\vec u+\vec v,\vec u-\vec v\right\rbrace\)",
+
+"\(\left\lbrace\frac{$normv}{$udotu\sqrt{$ssl}}\vec u+\frac{$normu}{$vdotv\sqrt{$ssl}}\vec v,\frac{$normu}{$udotu\sqrt{$ssl}}\vec u+\frac{$normv}{$vdotv\sqrt{$ssl}}\vec v\right\rbrace\)",
+    );
+$mc2 -> makeLast("None of the above");
+
+
+$mc3 = new_checkbox_multiple_choice();
+$mc3 -> qa (
+"Based on the previous two questions, which of the following sets are orthonormal? $BITALIC Select all that apply$EITALIC.", 
+
+"\(\left\lbrace\frac1{$normu}\vec u,\frac1{$normv}\vec v\right\rbrace\)",
+    );
+$mc3 -> extra(
+"\(\left\lbrace\vec u+\vec v,\vec u-\vec v\right\rbrace\)",
+
+"\(\left\lbrace\frac{$normv}{$udotu\sqrt{$ssl}}\vec u+\frac{$normu}{$vdotv\sqrt{$ssl}}\vec v,\frac{$normu}{$udotu\sqrt{$ssl}}\vec u+\frac{$normv}{$vdotv\sqrt{$ssl}}\vec v\right\rbrace\)",
+
+"\(\left\lbrace\frac1{$normu}\vec u+\frac1{$normv}\vec v,\frac1{$normu}\vec u-\frac1{$normv}\vec v\right\rbrace\)",
+    );
+$mc3 -> makeLast("None of the above");
+
+BEGIN_TEXT
+<p>
+Suppose \(\vec u\) and \(\vec v\) are vectors with \(\left\|\vec u\right\|=$normu\), \(\left\|\vec v\right\|=$normv\), and \(\vec u\cdot\vec v=$dotP\).
+</p>
+
+$BR
+
+\{ $mc1 -> print_q() \}
+
+$BR
+
+\{ $mc1 -> print_a() \}
+
+$BR
+$BR
+
+\{ $mc2 -> print_q() \}
+
+$BR
+
+\{ $mc2 -> print_a() \}
+
+$BR
+$BR
+
+\{ $mc3 -> print_q() \}
+
+$BR
+
+\{ $mc3 -> print_a() \}
+
+
+
+END_TEXT
+Context()->normalStrings;
+
+##################################################
+#  Answers
+
+$showPartialCorrectAnswers = 0;
+
+install_problem_grader(~~&std_problem_grader);
+
+$showPartialCorrectAnswers = 0;
+
+ANS(checkbox_cmp($mc1->correct_ans()));
+ANS(checkbox_cmp($mc2->correct_ans()));
+ANS(checkbox_cmp($mc3->correct_ans()));
+
+
+
+ENDDOCUMENT();

Contrib/GeorgiaTech/Module1TrueorFalse.pg

@@ -0,0 +1,150 @@
+## DBsubject(Calculus - multivariable)
+
+## DBchapter(Calculus of vector valued functions)
+
+## DBsection(Parameterized curves)
+
+## Institution(Georgia Institute of Technology)  
+
+## Author(Gregory Mayer, Hunter Lehman)  
+
+## Level(2)  
+
+## MO(1)
+
+## Language(en)
+
+
+DOCUMENT();      
+loadMacros(
+    "PGstandard.pl",
+    "MathObjects.pl",
+    "PGchoicemacros.pl",
+    "unionLists.pl",
+    "parserPopUp.pl",
+    "PGcourse.pl",
+);
+
+TEXT(beginproblem(), $BR,$BBOLD, "True or False Exercise", $EBOLD, $BR,$BR);
+$showPartialCorrectAnswers = 0;
+
+@questions = (
+    "The speed of a particle with a position function \(r(t)\) is \(\frac{r'(t)}{|r'(t)|}\).",
+    "If \(u\) and \(v\) are two vectors in \(\mathbb{R}^3\), then \(u \cdot v\) is a vector orthogonal to both \(u\) and \(v\).",
+    "The point (2, -1, 3) lies on the graph of the sphere \((x - 2)^2 + (y + 1)^2 + (z - 3)^2 = 25\).",
+    "\(\frac{d}{dt}[u(t) \times u(t)] = 2u'(t) \times u(t)\). ",
+    "Parallel planes have parallel normal vectors. ",
+    "The vector \(\frac{1}{\sqrt{3}} \hat i - \frac{1}{\sqrt{3}} \hat j + \frac{2}{\sqrt{3}} \hat k\) is a unit vector. ",
+    "The only way that \(v \cdot w\) = 0 is if \(v\) = 0 or \(w\) = 0.",
+);  
+
+@corans = ('False', 'False', 'False', 'False', 'True', 'True', 'False');
+
+# Choose Seven of the question and answer pairs at random.
+@which = NchooseK(7, 7);
+
+@squestions = @questions[@which];
+@scorans = @corans[@which];
+
+foreach my $i (0..6) {
+    $popup[$i] = PopUp(['Choose','True', 'False'],$scorans[$i]);
+}
+
+BEGIN_TEXT
+Indicate whether the following statements are true or false.
+$PAR
+\{ BeginList('OL',type=>'a') \}
+
+$ITEM
+$squestions[0]
+$BR
+\{ $popup[0]->menu \}
+$ITEMSEP
+
+$ITEM
+$squestions[1]
+$BR
+\{ $popup[1]->menu \}
+$ITEMSEP
+
+$ITEM
+$squestions[2]
+$BR
+\{ $popup[2]->menu \}
+$ITEMSEP
+
+$ITEM
+$squestions[3]
+$BR
+\{ $popup[3]->menu \}
+$ITEMSEP
+
+$ITEM
+$squestions[4]
+$BR
+\{ $popup[4]->menu \}
+$ITEMSEP
+
+$ITEM
+$squestions[5]
+$BR
+\{ $popup[5]->menu \}
+$ITEMSEP
+
+$ITEM
+$squestions[6]
+$BR
+\{ $popup[6]->menu \}
+$ITEMSEP
+
+\{ EndList('OL') \}
+
+END_TEXT
+
+install_problem_grader(~~&std_problem_grader);
+
+# Enter the correct answers to be checked against the answers to the students.
+
+foreach my $i (0..6) {
+    ANS($popup[$i]->cmp);
+}
+
+Context()->texStrings;
+
+@solutions = (
+    "$BBOLD False $EBOLD 
+    $BR
+    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. ",
+    "$BBOLD False $EBOLD 
+    $BR
+    \(u \cdot v\) 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.",
+    "$BBOLD False $EBOLD 
+    $BR
+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. ",
+    "$BBOLD False $EBOLD 
+    $BR
+    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.",
+     "$BBOLD True $EBOLD 
+     $BR
+     Parallel planes have identical normal vectors or normal vectors that are scalar multiples of each other. ",
+    "$BBOLD True $EBOLD 
+    $BR
+    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. ",
+    "$BBOLD False $EBOLD 
+    $BR
+    Two non-zero orthogonal vectors can have a dot product of zero.",
+);
+
+$solutions = "";
+foreach (@which) {
+    $solutions .= "$ITEM $solutions[$_]";
+}
+
+SOLUTION(EV3(<<'END_SOLUTION'));
+\{ BeginList('OL',type=>'a') \}
+$solutions
+\{ EndList('OL') \}
+END_SOLUTION
+Context()->normalStrings;
+
+ENDDOCUMENT();  # This should be the last executable line in the problem.