This shows how to use vectors in a problem.
Download file: Vectors.pg
DOCUMENT();
loadMacros('PGstandard.pl', 'PGML.pl', 'PGcourse.pl');
Preamble
These standard macros need to be loaded.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; # 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
We indicate that we are working in a vector context by setting Context('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. Any vector constructed with i, j and/or k will be three-dimensional vectors even if the vector k is not used. That is Vector('<1,2>') is not equal to i+2j.
Also if we define the vector using the constants i, j and k, as in the definition of $v3here, then the default display of that vector will be in i,j,kformat 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 MathObjects vectors are available as an array from $v->value; thus, we could save the three components of the vector $v3 in the array @v3comp using @v3comp = $v3->value. Then, we can access the first component using $v3comp[0], the second component using $v3comp[1], etc. Better still, to get the first component of the vector $v3 we could use $v3->extract(1) instead of ($v3->value)[0]. Note that the index of extract starts at 1, not 0.
Lastly, there is other functionality associated with Vectors. See the WeBWorK wiki page on vectors
BEGIN_PGML_SOLUTION Solution explanation goes here. END_PGML_SOLUTION ENDDOCUMENT();
Solution
A solution should be provided here.