Difference between revisions of "Vectors"

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Line 51: Line 51:
 
$v2 = Compute("<-3,1>");
 
$v2 = Compute("<-3,1>");
 
$v3 = 3*i + 2*j - 4*k;
 
$v3 = 3*i + 2*j - 4*k;
  +
$v4 = Vector(1,1,0);
   
 
# create an array of the components of $v3
 
# create an array of the components of $v3
Line 57: Line 58:
 
$a = 3*i + j;
 
$a = 3*i + j;
 
$b = $a + $v1;
 
$b = $a + $v1;
  +
  +
$k = norm($v3); # vector length
  +
$v5 = unit($v3); # unit vector in same direction
  +
$c = $v1 . $v2; # dot product
  +
$v6 = $v3 x $v4; # cross product
  +
$v3->isParallel($v4); # =1 if parallel, =0 if skew
 
</pre>
 
</pre>
 
</td>
 
</td>

Revision as of 23:46, 27 October 2009

Vectors in Problems: PG Code Snippet

This code snippet shows the essential PG code to use vectors in WeBWorK problems. Note that these are insertions, not a complete PG file. This code will have to be incorporated into the problem file on which you are working.

Problem Techniques Index

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

Be sure to load MathObjects.pl and parserVectorUtils.pl

  Context('Vector');
  ## display vectors in ijk format
  # Context()->flags->set( ijk=>1 );

  ## set the appearance of the ijk vectors
  ##    this sets them to be overset with
  ##    vector arrows, instead of boldface
  # Context()->constants->set(
  #   i => {TeX => "\mathit{\vec i}"},
  #   j => {TeX => "\mathit{\vec j}"},
  #   k => {TeX => "\mathit{\vec k}"},
  # );

  $v1 = Vector("<1,3>");
  $v2 = Compute("<-3,1>");
  $v3 = 3*i + 2*j - 4*k;
  $v4 = Vector(1,1,0);

  # create an array of the components of $v3
  @v3comp = $v3->value;

  $a = 3*i + j;
  $b = $a + $v1;

  $k = norm($v3);       # vector length   
  $v5 = unit($v3);      # unit vector in same direction
  $c = $v1 . $v2;       # dot product
  $v6 = $v3 x $v4;      # cross product
  $v3->isParallel($v4); # =1 if parallel, =0 if skew

We indicate that we are working in a vector context by setting the Context to be Vector. If we want to have vectors displayed, by default, as a sum of i,j,k components, we can set the ijk flag in the Context. This is commented out here; uncommenting it would result in the vector v1 here being shown as v1 = i + 3j instead of v1 = <1,3>, etc. Similarly, if we wanted to change the default display of the i, j and k vectors, say to have them display with overset arrows, we can redefine the TeX formatting of those constants, as shown in the second comment.

Then, we can define vectors as we might expect: either with the Vector or Compute constructors, or by using the predefined vector constants i, j and k.

Note that if we define the vector using the constants i, j and k, as in the definition of $v3 here, then the default display of that vector will be in i,j,k format even if we don't set the corresponding Context flag.

To explicitly require that the vectors be two-dimensional rather than three-dimensional, we would use Context('Vector2D') instead of Context('Vector').

The components of the vectors are available as an array from $v->value; thus, we could save the three components of the vector $v3 in the array @v3comp. Then, we can access the first component using $v3comp[0], the second component using $v3comp[1], etc.

  BEGIN_TEXT
  Enter the vector pointing from
  \($a\) to \($b\):
  \{ ans_rule(25) \}
  $PAR
  Enter a vector perpendicular to this:
  \{ ans_rule(25) \}
  $PAR
  Enter a vector parallel to \($v3\):
  \{ ans_rule(25) \}

We can then use the vectors that we created in the text section of the problem.

  ANS( $v1->cmp() );

  ANS( $v2->cmp( checker=>sub {
    my ($correct, $student, $ansHash) = @_;
    return $student->isParallel($correct);
  }, showCoordinateHints => 0 ) );
  ## or:
  # ANS( $v1->cmp( checker=>sub {
  #   my ($correct, $student, $ansHash) = @_;
  #   return $student.$correct == 0; } ) );

  # make a custom answer checker as a subroutine
  sub parallel_vector_cmp {
    my ($correct, $student, $ansHash) = @_;
    return $student->isParallel($correct);
  }
  ANS( $v3->cmp( checker=>~~&parallel_vector_cmp, showCoordinateHints => 0 ) );

We can then use the vectors to check the answers that are given. Note that we have used custom answer checkers for the latter answer evaluators here, taking advantage of the built in dot product and isParallel method of vector objects. When checking if a student's vector is parallel to the correct vector, hints about which coordinates are incorrect can be misleading, so we set showCoordinateHints => 0.

Other properties of MathObjects vectors are given in the MathObjects reference table.

Problem Techniques Index