Difference between revisions of "ParametricLineAnswers"

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  +
{{historical}}
  +
  +
<p style="font-size: 120%;font-weight:bold">This problem has been replaced with many other problems:</p>
  +
* [https://openwebwork.github.io/pg-docs/sample-problems/VectorCalc/VectorParametricLine.html A parametric line]
  +
* [https://openwebwork.github.io/pg-docs/sample-problems/VectorCalc/VectorParametricLine.html A vector-valued parametric line]
  +
* [https://openwebwork.github.io/pg-docs/sample-problems/VectorCalc/VectorLineSegment1.html Vector-valued line segment--general]
  +
* [https://openwebwork.github.io/pg-docs/sample-problems/VectorCalc/VectorLineSegment2.html Vector-valued line segment--specific]
  +
  +
<h2>Parametric Lines as Answers</h2>
  +
  +
<!-- Header for these sections -- no modification needed -->
  +
  +
<p style="background-color:#eeeeee;border:black solid 1px;padding:3px;">
  +
<em>This shows the PG code to check student answers that are vectors whose components are formulas.</em>
  +
</p>
  +
<ul>
  +
<li><b>Example 1:</b> A vector parametric line where the student chooses the parametrization</li>
  +
<li><b>Example 2:</b> A vector parametric line segment where the student chooses the parametrization and the starting and ending times</li>
  +
<li><b>Example 3:</b> A vector parametric line segment where the student must enter a particular parametrization and use particular starting and ending times</li>
  +
</ul>
  +
  +
  +
  +
  +
<p style="background-color:#eeeeee;border:black solid 1px;padding:3px;">
  +
<em><b>Example 1:</b> A vector parametric line where the student chooses the parametrization</em>
  +
</p>
  +
  +
<p style="text-align:center;">
  +
[[IndexOfProblemTechniques|Problem Techniques Index]]
  +
</p>
  +
  +
<table cellspacing="0" cellpadding="2" border="0">
  +
  +
<tr valign="top">
  +
<th> PG problem file </th>
  +
<th> Explanation </th>
  +
</tr>
  +
  +
<!-- Load specialized macro files section -->
  +
  +
<tr valign="top">
  +
<td style="background-color:#ddffdd;border:black 1px dashed;">
  +
<pre>
  +
DOCUMENT();
  +
loadMacros(
  +
"PGstandard.pl",
  +
"MathObjects.pl",
  +
"parserVectorUtils.pl",
  +
"parserParametricLine.pl",
  +
"PGcourse.pl",
  +
);
  +
TEXT(beginproblem());
  +
</pre>
  +
</td>
  +
<td style="background-color:#ccffcc;padding:7px;">
  +
<p>
  +
<b>Initialization:</b>
  +
The first three macros should always be loaded for questions whose answers are vector valued functions.
  +
For the general vector parametric line we need to load <code>parserParametricLine.pl</code>.
  +
</p>
  +
</td>
  +
</tr>
  +
  +
<!-- Setup section -->
  +
  +
<tr valign="top">
  +
<td style="background-color:#ffffdd;border:black 1px dashed;">
  +
<pre>
  +
Context("Vector");
  +
Context()->variables->are(t=>"Real");
  +
  +
$P = non_zero_point3D();
  +
$disp = non_zero_vector3D();
  +
$Q = Point($P + $disp);
  +
</pre>
  +
</td>
  +
<td style="background-color:#ffffcc;padding:7px;">
  +
<p>
  +
<b>Setup:</b>
  +
We randomize two points in three-dimensional space, P and Q, a displacement vector between them, and a speed to travel between them.
  +
</p>
  +
</td>
  +
</tr>
  +
  +
<!-- Question text section -->
  +
  +
<tr valign="top">
  +
<td style="background-color:#ffdddd;border:black 1px dashed;">
  +
<pre>
  +
Context()->texStrings;
  +
BEGIN_TEXT
  +
Find a vector parametric equation for the
  +
line through points \( P = $P \) and \( Q = $Q \).
  +
$BR
  +
\( \vec{r}(t) = \) \{ ans_rule(40) \}
  +
END_TEXT
  +
Context()->normalStrings;
  +
</pre>
  +
<td style="background-color:#ffcccc;padding:7px;">
  +
<p>
  +
<b>Main Text:</b>
  +
The problem text section of the file is as we'd expect.
  +
</p>
  +
</td>
  +
</tr>
  +
  +
<!-- Answer section -->
  +
  +
<tr valign="top">
  +
<td style="background-color:#eeddff;border:black 1px dashed;">
  +
<pre>
  +
$showPartialCorrectAnswers = 1;
  +
  +
ANS( ParametricLine("$P + t * $disp")->cmp() );
  +
  +
ENDDOCUMENT();
  +
</pre>
  +
<td style="background-color:#eeccff;padding:7px;">
  +
<p>
  +
<b>Answer Evaluation:</b>
  +
The answer can be any vector parametric line through the points P and Q, so we use <code>ParametricLine("$P + t * $disp")</code> provided by <code>parserParametricLine.pl</code> to allow student answers that use a different parametrization, such as <code>$Q - 2*t * $disp</code>, to be marked correct.
  +
</p>
  +
</td>
  +
</tr>
  +
</table>
  +
  +
<p style="text-align:center;">
  +
[[IndexOfProblemTechniques|Problem Techniques Index]]
  +
</p>
  +
  +
[[Category:Problem Techniques]]
  +
  +
  +
  +
  +
  +
  +
<p style="background-color:#eeeeee;border:black solid 1px;padding:3px;">
  +
<em><b>Example 2:</b> A vector parametric line segment where the student chooses the parametrization and the starting and ending times</em>
  +
</p>
  +
  +
<p style="text-align:center;">
  +
[[IndexOfProblemTechniques|Problem Techniques Index]]
  +
</p>
  +
  +
<table cellspacing="0" cellpadding="2" border="0">
  +
  +
<tr valign="top">
  +
<th> PG problem file </th>
  +
<th> Explanation </th>
  +
</tr>
  +
  +
<!-- Load specialized macro files section -->
  +
  +
<tr valign="top">
  +
<td style="background-color:#ddffdd;border:black 1px dashed;">
  +
<pre>
  +
DOCUMENT();
  +
loadMacros(
  +
"PGstandard.pl",
  +
"MathObjects.pl",
  +
"parserVectorUtils.pl",
  +
"parserParametricLine.pl",
  +
"parserMultiAnswer.pl",
  +
"PGcourse.pl",
  +
);
  +
TEXT(beginproblem());
  +
</pre>
  +
</td>
  +
<td style="background-color:#ccffcc;padding:7px;">
  +
<p>
  +
<b>Initialization:</b>
  +
The first three macros should always be loaded for questions whose answers are vector valued functions.
  +
We need to load <code>parserParametricLine.pl</code> and <code>parserMultiAnswer.pl</code> since we will want to evaluate the student's answer at the starting and ending times provided by the student.
  +
</p>
  +
</td>
  +
</tr>
  +
  +
<!-- Setup section -->
  +
  +
<tr valign="top">
  +
<td style="background-color:#ffffdd;border:black 1px dashed;">
  +
<pre>
  +
Context("Vector");
  +
Context()->variables->are(t=>"Real");
  +
  +
$P = Point(4,0);
  +
$Q = Point(0,2);
  +
$V = Vector(-4,2);
  +
  +
$t = Formula("t");
  +
$line = Vector("$P + $t * $V");
  +
  +
$multians = MultiAnswer( $line, Real("0"), Real("1") )->with(
  +
singleResult => 0,
  +
checker => sub {
  +
my ( $correct, $student, $ansHash ) = @_;
  +
my ( $linestu, $astu, $bstu ) = @{$student};
  +
my ( $linecor, $acor, $bcor ) = @{$correct};
  +
  +
if ( (ParametricLine("$line") == $linestu) &&
  +
($linestu->eval(t=>"$astu") == $line->eval(t=>"0")) &&
  +
($linestu->eval(t=>"$bstu") == $line->eval(t=>"1")) )
  +
{
  +
return [1,1,1];
  +
  +
} elsif ( (ParametricLine("$line") == $linestu) &&
  +
($linestu->eval(t=>"$astu") == $line->eval(t=>"0")) )
  +
{
  +
return [1,1,0];
  +
  +
} elsif ( (ParametricLine("$line") == $linestu) &&
  +
($linestu->eval(t=>"$bstu") == $line->eval(t=>"1")) )
  +
{
  +
return [1,0,1];
  +
  +
} elsif ( (ParametricLine("$line") == $linestu) )
  +
{
  +
return [1,0,0];
  +
  +
} else {
  +
return [0,0,0];
  +
}
  +
  +
}
  +
);
  +
</pre>
  +
</td>
  +
<td style="background-color:#ffffcc;padding:7px;">
  +
<p>
  +
<b>Setup:</b>
  +
We create a MutiAnswer answer checker that will evaluate the students vector parametric equation at the starting and ending times provided by the student. For example, both of the student answers <code>(4,0) + t<-4,2></code> for t between 0 and 1, and <code>(4,0) + t<-2,1></code> for t between 0 and 2 will be marked correct.
  +
</p>
  +
</td>
  +
</tr>
  +
  +
<!-- Question text section -->
  +
  +
<tr valign="top">
  +
<td style="background-color:#ffdddd;border:black 1px dashed;">
  +
<pre>
  +
Context()->texStrings;
  +
BEGIN_TEXT
  +
Find a vector parametric equation for the line
  +
segment from the point \(P = $P\)
  +
to \(Q = $Q\).
  +
$BR
  +
\( \vec{r}(t) = \)
  +
\{ $multians->ans_rule(40 )\}
  +
for
  +
\{ $multians->ans_rule(5) \}
  +
\( \leq t \leq \)
  +
\{ $multians->ans_rule(5) \}
  +
END_TEXT
  +
Context()->normalStrings;
  +
</pre>
  +
<td style="background-color:#ffcccc;padding:7px;">
  +
<p>
  +
<b>Main Text:</b>
  +
The problem text section of the file is as we'd expect.
  +
</p>
  +
</td>
  +
</tr>
  +
  +
<!-- Answer section -->
  +
  +
<tr valign="top">
  +
<td style="background-color:#eeddff;border:black 1px dashed;">
  +
<pre>
  +
$showPartialCorrectAnswers = 1;
  +
  +
ANS( $multians->cmp() );
  +
  +
ENDDOCUMENT();
  +
</pre>
  +
<td style="background-color:#eeccff;padding:7px;">
  +
<p>
  +
<b>Answer Evaluation:</b>
  +
This part is easy since we defined <code>$multians</code> above.
  +
</p>
  +
</td>
  +
</tr>
  +
</table>
  +
  +
<p style="text-align:center;">
  +
[[IndexOfProblemTechniques|Problem Techniques Index]]
  +
</p>
  +
  +
[[Category:Problem Techniques]]
  +
  +
  +
  +
  +
  +
  +
  +
<p style="background-color:#eeeeee;border:black solid 1px;padding:3px;">
  +
<em><b>Example 3:</b> A vector parametric line segment where the student must enter a particular parametrization and use particular starting and ending times</em>
  +
</p>
  +
  +
<p style="text-align:center;">
  +
[[IndexOfProblemTechniques|Problem Techniques Index]]
  +
</p>
  +
  +
<table cellspacing="0" cellpadding="2" border="0">
  +
  +
<tr valign="top">
  +
<th> PG problem file </th>
  +
<th> Explanation </th>
  +
</tr>
  +
  +
<!-- Load specialized macro files section -->
  +
  +
<tr valign="top">
  +
<td style="background-color:#ddffdd;border:black 1px dashed;">
  +
<pre>
  +
DOCUMENT();
  +
loadMacros(
  +
"PGstandard.pl",
  +
"MathObjects.pl",
  +
"parserVectorUtils.pl",
  +
"answerCustom.pl",
  +
"PGcourse.pl",
  +
);
  +
TEXT(beginproblem());
  +
</pre>
  +
</td>
  +
<td style="background-color:#ccffcc;padding:7px;">
  +
<p>
  +
<b>Initialization:</b>
  +
The first three macros should always be loaded for questions whose answers are vector valued functions.
  +
For the specific vector parametric line (part (b) in the question below) we need to load <code>answerCustom.pl</code> since we will use a custom answer checker and want to have features like <code>showCoordinateHints</code> enabled.
  +
</p>
  +
</td>
  +
</tr>
  +
  +
<!-- Setup section -->
  +
  +
<tr valign="top">
  +
<td style="background-color:#ffffdd;border:black 1px dashed;">
  +
<pre>
  +
Context("Vector");
  +
Context()->variables->are(t=>"Real");
  +
  +
$P = non_zero_point3D();
  +
$disp = non_zero_vector3D();
  +
$Q = Point($P + $disp);
  +
$speed = random(3,9,1);
  +
</pre>
  +
</td>
  +
<td style="background-color:#ffffcc;padding:7px;">
  +
<p>
  +
<b>Setup:</b>
  +
We randomize two points in three-dimensional space, P and Q, a displacement vector between them, and a speed to travel between them.
  +
</p>
  +
</td>
  +
</tr>
  +
  +
<!-- Question text section -->
  +
  +
<tr valign="top">
  +
<td style="background-color:#ffdddd;border:black 1px dashed;">
  +
<pre>
  +
Context()->texStrings;
  +
BEGIN_TEXT
  +
A particle starts at the point \( P = $P \)
  +
when \( t = 0 \) and moves along a straight line
  +
toward \( Q = $Q \) at a speed of \( $speed \)
  +
cm/sec. Assume that x, y, and z are measured
  +
in cm. Do not enter units with your answers.
  +
$BR
  +
$BR
  +
Find the vector parametric equation
  +
for the position of the object.
  +
$BR
  +
\( \vec{r}(t) = \) \{ans_rule(40)\}
  +
END_TEXT
  +
Context()->normalStrings;
  +
</pre>
  +
<td style="background-color:#ffcccc;padding:7px;">
  +
<p>
  +
<b>Main Text:</b>
  +
The problem text section of the file is as we'd expect.
  +
</p>
  +
</td>
  +
</tr>
  +
  +
<!-- Answer section -->
  +
  +
<tr valign="top">
  +
<td style="background-color:#eeddff;border:black 1px dashed;">
  +
<pre>
  +
$showPartialCorrectAnswers = 1;
  +
  +
# for checking a particular vector parametric line
  +
sub mycheck {
  +
my ($correct, $student, $ansHash) = @_;
  +
  +
if (
  +
($correct . i == $student . i) &&
  +
($correct . j == $student . j) &&
  +
($correct . k == $student . k)
  +
)
  +
{ return 1; } else { return 0; }
  +
}
  +
  +
$T = Formula("$speed * t / norm($disp)");
  +
$r = $P + $T * $disp;
  +
  +
ANS( custom_cmp( $r, ~~&mycheck, showCoordinateHints=>1 ) );
  +
  +
ENDDOCUMENT();
  +
</pre>
  +
<td style="background-color:#eeccff;padding:7px;">
  +
<p>
  +
<b>Answer Evaluation:</b>
  +
There is only one vector parametric equation that describes the position of the object subject to the given conditions, so we want only one answer to be marked correct. We create a custom answer checker <code>mycheck</code>, which is a subroutine that verifies that the student's answer agrees with the correct answer in each component by taking the dot product with the each of the vectors i, j, and k separately. We use <code>custom_cmp( $correct_ans, ~~&custom_checker, %options )</code> provided by the macro <code>answerCustom.pl</code> so that we can enable the <code>showCoordinateHints=>1</code> feature.
  +
</p>
  +
<p>
  +
For more on custom answer evaluators, see [http://webwork.maa.org/wiki/CustomAnswerCheckers CustomAnswerCheckers] and
  +
[http://webwork.maa.org/doc/cvs/pg_CURRENT/macros/answerCustom.pl.html answerCustom.pl.html]
  +
</p>
  +
</td>
  +
</tr>
  +
</table>
  +
  +
<p style="text-align:center;">
  +
[[IndexOfProblemTechniques|Problem Techniques Index]]
  +
</p>
  +
  +
[[Category:Problem Techniques]]
  +
  +
  +
  +
  +
  +
  +
<ul>
  +
<li>POD documentation: [http://webwork.maa.org/pod/pg/macros/parserParametricLine.html parserParametricLine.pl]</li>
  +
<li>PG macro: [http://webwork.maa.org/viewvc/system/trunk/pg/macros/parserParametricLine.pl?view=log parserParametricLine.pl]</li>
  +
</ul>
  +
  +
  +
  +
<ul>
  +
<li>POD documentation: [http://webwork.maa.org/pod/pg/macros/answerCustom.html answerCustom.pl]</li>
  +
<li>PG macro: [http://webwork.maa.org/viewvc/system/trunk/pg/macros/answerCustom.pl?view=log answerCustom.pl]</li>
  +
</ul>

Latest revision as of 11:04, 16 July 2023

This article has been retained as a historical document. It is not up-to-date and the formatting may be lacking. Use the information herein with caution.

This problem has been replaced with many other problems:

Parametric Lines as Answers


This shows the PG code to check student answers that are vectors whose components are formulas.

  • Example 1: A vector parametric line where the student chooses the parametrization
  • Example 2: A vector parametric line segment where the student chooses the parametrization and the starting and ending times
  • Example 3: A vector parametric line segment where the student must enter a particular parametrization and use particular starting and ending times



Example 1: A vector parametric line where the student chooses the parametrization

Problem Techniques Index

PG problem file Explanation 
DOCUMENT();
loadMacros(
"PGstandard.pl",
"MathObjects.pl",
"parserVectorUtils.pl",
"parserParametricLine.pl",
"PGcourse.pl",
);
TEXT(beginproblem());

Initialization: The first three macros should always be loaded for questions whose answers are vector valued functions. For the general vector parametric line we need to load parserParametricLine.pl. 

Context("Vector");
Context()->variables->are(t=>"Real");

$P = non_zero_point3D();
$disp = non_zero_vector3D();
$Q = Point($P + $disp);

Setup: We randomize two points in three-dimensional space, P and Q, a displacement vector between them, and a speed to travel between them. 

Context()->texStrings;
BEGIN_TEXT
Find a vector parametric equation for the 
line through points \( P = $P \) and \( Q = $Q \).
$BR
\( \vec{r}(t) = \) \{ ans_rule(40) \}
END_TEXT
Context()->normalStrings;

Main Text: The problem text section of the file is as we'd expect. 

$showPartialCorrectAnswers = 1;

ANS( ParametricLine("$P + t * $disp")->cmp() );

ENDDOCUMENT();

Answer Evaluation: The answer can be any vector parametric line through the points P and Q, so we use ParametricLine("$P + t * $disp") provided by parserParametricLine.pl to allow student answers that use a different parametrization, such as $Q - 2*t * $disp, to be marked correct.

Problem Techniques Index




Example 2: A vector parametric line segment where the student chooses the parametrization and the starting and ending times

Problem Techniques Index

PG problem file Explanation 
DOCUMENT();
loadMacros(
"PGstandard.pl",
"MathObjects.pl",
"parserVectorUtils.pl",
"parserParametricLine.pl",
"parserMultiAnswer.pl",
"PGcourse.pl",
);
TEXT(beginproblem());

Initialization: The first three macros should always be loaded for questions whose answers are vector valued functions. We need to load parserParametricLine.pl and parserMultiAnswer.pl since we will want to evaluate the student's answer at the starting and ending times provided by the student. 

Context("Vector");
Context()->variables->are(t=>"Real");

$P = Point(4,0);
$Q = Point(0,2);
$V = Vector(-4,2);

$t = Formula("t");
$line = Vector("$P + $t * $V");

$multians = MultiAnswer( $line, Real("0"), Real("1") )->with(
  singleResult => 0,
  checker => sub {
    my ( $correct, $student, $ansHash ) = @_;
    my ( $linestu, $astu, $bstu ) = @{$student};
    my ( $linecor, $acor, $bcor ) = @{$correct};

    if ( (ParametricLine("$line") == $linestu) &&
         ($linestu->eval(t=>"$astu") == $line->eval(t=>"0")) &&
         ($linestu->eval(t=>"$bstu") == $line->eval(t=>"1")) ) 
    {  
       return [1,1,1]; 

    } elsif ( (ParametricLine("$line") == $linestu) &&
              ($linestu->eval(t=>"$astu") == $line->eval(t=>"0")) )
    {  
       return [1,1,0]; 

    } elsif ( (ParametricLine("$line") == $linestu) &&
              ($linestu->eval(t=>"$bstu") == $line->eval(t=>"1")) )
    {  
       return [1,0,1]; 

    } elsif ( (ParametricLine("$line") == $linestu) )
    {
        return [1,0,0];

    } else { 
        return [0,0,0]; 
    }

  }
);

Setup: We create a MutiAnswer answer checker that will evaluate the students vector parametric equation at the starting and ending times provided by the student. For example, both of the student answers (4,0) + t<-4,2> for t between 0 and 1, and (4,0) + t<-2,1> for t between 0 and 2 will be marked correct. 

Context()->texStrings;
BEGIN_TEXT
Find a vector parametric equation for the line 
segment from the point \(P = $P\) 
to \(Q = $Q\).  
$BR
\( \vec{r}(t) = \) 
\{ $multians->ans_rule(40 )\}
for 
\{ $multians->ans_rule(5) \}
\( \leq t \leq \)
\{ $multians->ans_rule(5) \}
END_TEXT
Context()->normalStrings;

Main Text: The problem text section of the file is as we'd expect. 

$showPartialCorrectAnswers = 1;

ANS( $multians->cmp() );

ENDDOCUMENT();

Answer Evaluation: This part is easy since we defined $multians above.

Problem Techniques Index




Example 3: A vector parametric line segment where the student must enter a particular parametrization and use particular starting and ending times

Problem Techniques Index

PG problem file Explanation 
DOCUMENT();
loadMacros(
"PGstandard.pl",
"MathObjects.pl",
"parserVectorUtils.pl",
"answerCustom.pl",
"PGcourse.pl",
);
TEXT(beginproblem());

Initialization: The first three macros should always be loaded for questions whose answers are vector valued functions. For the specific vector parametric line (part (b) in the question below) we need to load answerCustom.pl since we will use a custom answer checker and want to have features like showCoordinateHints enabled. 

Context("Vector");
Context()->variables->are(t=>"Real");

$P = non_zero_point3D();
$disp = non_zero_vector3D();
$Q = Point($P + $disp);
$speed = random(3,9,1);

Setup: We randomize two points in three-dimensional space, P and Q, a displacement vector between them, and a speed to travel between them. 

Context()->texStrings;
BEGIN_TEXT
A particle starts at the point \( P = $P \)
when \( t = 0 \) and moves along a straight line
toward \( Q = $Q \) at a speed of \( $speed \) 
cm/sec.  Assume that x, y, and z are measured
in cm.  Do not enter units with your answers.
$BR
$BR
Find the vector parametric equation
for the position of the object.
$BR
\( \vec{r}(t) = \) \{ans_rule(40)\}
END_TEXT
Context()->normalStrings;

Main Text: The problem text section of the file is as we'd expect. 

$showPartialCorrectAnswers = 1;

#  for checking a particular vector parametric line
sub mycheck {
  my ($correct, $student, $ansHash) = @_;

  if ( 
       ($correct . i == $student . i) && 
       ($correct . j == $student . j) && 
       ($correct . k == $student . k) 
     ) 
  { return 1; } else { return 0; }
}

$T = Formula("$speed * t / norm($disp)");
$r = $P + $T * $disp;

ANS( custom_cmp( $r, ~~&mycheck, showCoordinateHints=>1 ) );

ENDDOCUMENT();

Answer Evaluation: There is only one vector parametric equation that describes the position of the object subject to the given conditions, so we want only one answer to be marked correct. We create a custom answer checker mycheck, which is a subroutine that verifies that the student's answer agrees with the correct answer in each component by taking the dot product with the each of the vectors i, j, and k separately. We use custom_cmp( $correct_ans, ~~&custom_checker, %options ) provided by the macro answerCustom.pl so that we can enable the showCoordinateHints=>1 feature.

For more on custom answer evaluators, see CustomAnswerCheckers and answerCustom.pl.html

Problem Techniques Index





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