VectorParametricLines

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This article has been retained as a historical document.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 a parametric line problem, a vector line segment problem and another vector line segment problem

Vector 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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