Vectors.pg

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

# Uncomment to change the appearance of the ijk vectors. This sets them to be
# overset with vector arrows, instead of bold.
#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; # this results in an error
$c  = norm($v3);         # vector length
$v5 = unit($v3);         # unit vector in same direction
$d  = $v1 . $v2;         # dot product
$v6 = $v3 x $v4;         # cross product
$v3->isParallel($v4);    # returns 1 if parallel, 0 if skew

BEGIN_PGML
[`\vec{v}_1 = [$v1]`]

[`\vec{v}_2 = [$v2]`]

[`\vec{v}_3 = [$v3]`]

[`\vec{v}_4 = [$v4]`]

[`\vec{a} = [$a]`]

[`\vec{v}_1 + \vec{v}_2 = [$v1 + $v2]`]

[`||\vec{v}_3|| = [$c]`]

[`[$v5]`] is a unit vector in the same direction as [`[$v3]`]

[`\vec{v}_1 \cdot \vec{v}_2 = [$d]`]

[`\vec{v}_3 \times \vec{v}_4 = [$v6]`]

[`\vec{v}_3`] [@ $v3->isParallel($v4) ? 'is' : 'is not' @] parallel to
[`\vec{v}_4`]

The first element of [`\vec{v}_3`] is [@ $v3->extract(1) @]

The third element of [`\vec{v}_4`] is [@ $v4->extract(3) @]
END_PGML

Setup

Select the Vector context to work with vectors.

If vectors are to be displayed as a sum of i, j, k components, we can set the ijk context flag. This would result in the vector $v1 in this example being shown as $v1 = i + 3j instead of $v1 = <1, 3>, for example.

If it is desired to change how the i, j, and k vectors are displayed, say to have them displayed with over set arrows instead of bold, the TeX formatting of those constants can be redefined as shown in the second comment.

Vectors are defined either with the Vector or Compute constructors, or by using the predefined vector constants i, j and k. Any vector constructed with i, j, and k will be a three-dimensional vector even if the vector k is not used. That means Vector('<1, 2>') is not equal to i + 2j.

If a vector is defined using the constants i, j and k, as in the definition of $v3, then the default display of that vector will be in i, j, k format even if the ijk context flag is not set.

To explicitly require that the vectors be two-dimensional rather than three-dimensional, use the Vector2D context instead of the Vector context.

The components of vectors are available as an array returned by $v->value. Thus, the three components of the vector $v3 can be stored in the array @v3comp by calling @v3comp = $v3->value. The first component can then be accessed using $v3comp[0], the second component using $v3comp[1], and the third component using $v3comp[2].

To obtain a single component of a vector use the extract method. For example, to obtain the first component of $v3 use $v3->extract(1). Note that the index of extract starts at 1, not 0.

See the vector wiki page for more information about MathObject vectors.

Vectors

Complete Code

PG problem file

Explanation