Merge pull request #1168 from gajennings/main

Fix bug 4836, convert to PGML, add solution.

Author: gajennings

OpenProblemLibrary/Union/setMVvectors/an12_3_22.pg

@@ -23,8 +23,8 @@ DOCUMENT();        # This should be the first executable line in the problem.
 
 loadMacros(
   "PGstandard.pl",
-  "PGunion.pl",
   "MathObjects.pl",
+  "PGML.pl",
   "parserVectorUtils.pl",
   "PGcourse.pl"
 );
@@ -46,39 +46,44 @@ $V = non_zero_vector3D();
 #
 #  The projections
 #
-$U1 = (($U.$V)/($V.$V)) * $V;
-$U2 = $U - $U1;
+$U1 = Compute("(($U.$V)/($V.$V)) * $V");
+$U2 = Compute("$U - $U1");
+
+$u = Overline('u');
+$v = Overline('v');
+$w = Overline('w');
+
 
 ##############################################
 #  Main text
 
-$u = Overline('u');
-$v = Overline('v');
+BEGIN_PGML
+
+Suppose [`[$u] = [$U]`] and [`[$v] = [$V]`].  Then:
+
+a) The vector projection of [`[$u]`] along [`[$v]`] is [_]{$U1}{40}.  
+
+b) The vector projection of [`[$u]`] orthogonal to [`[$v]`] is [_]{$U2}{40}.
 
-Context()->texStrings;
-BEGIN_TEXT
+END_PGML
 
-Suppose \($u = $U\) and \($v = $V\).  Then:
+BEGIN_PGML_SOLUTION
+a) Let [``[$w] = \frac{[$v]}{\sqrt{[$v] \cdot [$v]}}``] be the unit vector that points in the same direction as [`[$u]`].  The _scalar projection_ of [`[$u]`] along [`[$v]`] is the dot product [``[$u]\cdot [$w] = \frac{[$u]\cdot [$v]}{\sqrt{[$v] \cdot [$v]}}``].  The _vector projection_ of [`[$u]`] along [`[$v]`] is the scalar projection times times [`[$w]`]:    
 
-\{BeginList()\}
-$ITEM
-The projection of \($u\) along \($v\) is \{ans_rule(40)\}.
-$ITEMSEP
+        [``\text{proj}_{[$v]}([$u]) =  \frac{[$u]\cdot [$v]}{\sqrt{[$v] \cdot [$v]}}\frac{[$v]}{\sqrt{[$v] \cdot [$v]}}= \frac{[$u]\cdot [$v]}{[$v] \cdot [$v]}[$v] = \frac{[@ $U.$V @]}{ [@ $V.$V @] }[$V]``]
 
-$ITEM
-The projection of \($u\) orthogonal to \($v\) is \{ans_rule(40)\}.
-\{EndList()\}
+b) [`[$v]`] is the sum of two perpendicular vectors, one points in the same direction as [`[$u]`], and the other points in a direction perpendicular to [`[$u]`].  The first is [``\text{proj}_{[$v]}([$u])``], the vector from part a). The other, [``[$v]-\text{proj}_{[$v]}([$u])``], is the vector that's requested in part b).  So the answer for part b) is    
 
-END_TEXT
-Context()->normalStrings;
+        [``[$U] - \frac{[@ $U.$V @]}{ [@ $V.$V @] }[$V]``]    
+END_PGML_SOLUTION
 
 ##################################################
 #  Answers
 
-ANS($U1->cmp);
-ANS($U2->cmp);
+#ANS($U1->cmp);
+#ANS($U2->cmp);
 
-$showPartialCorrectAnswers = 1;
+#$showPartialCorrectAnswers = 1;
 
 ##################################################