Vector Operations

Vector Calculus: Basic vector operations

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Complete Code

Download file: VectorOperations.pg

POD for Macro Files

PG problem file

Explanation

DOCUMENT();

loadMacros('PGstandard.pl', 'PGML.pl', 'parserVectorUtils.pl', 'PGcourse.pl');

Preamble

Load parserVectorUtils.pl for the non_zero_vector3D method.

Context('Vector');

$U = non_zero_vector3D(-9, 9, 1);
$V = non_zero_vector3D(-9, 9, 1);

# value works only for vectors of constants
@Uarray = $U->value;
$Ucomp2 = $Uarray[1];

$UdotV   = $U . $V;
$UcrossV = $U x $V;
$Vlength = norm($V);
$Vunit   = unit($V);

# Prevent students from entering the dot and cross products,
# and using the vector functions such as norm and unit.
Context()->operators->undefine('.', '><');
Context()->functions->disable('Vector');

BEGIN_PGML
Suppose [`\vec{u} = [$U]`] and [`\vec{v} = [$V]`].

a. The second component of [`\vec{u}`] is [_]{$Ucomp2}{5}

b. [`\vec{u} \cdot \vec{v} =`] [_]{$UdotV}{5}

c. [`\vec{u} \times \vec{v} =`] [_]{$UcrossV}{15}

d. [`\left\|\vec{v}\right\| =`] [_]{$Vlength}{10}

e. Enter a unit vector in the direction of [`\vec{v}`].  
   [_]{$Vunit}{15}

f. Enter a vector parallel to [`\vec{v}`].  
   [_]{$V->cmp(parallel => 1)}{15}

g. Enter a vector in the same direction as [`\vec{v}`].  
   [_]{$V->cmp( parallel => 1, sameDirection => 1 )}{15}
END_PGML

Setup

Use non_zero_vector3D(low, high, increment) to randomly generate two vectors.

Calling $U->value returns a (Perl) array of numbers which are the components of the vector $U. (Note that ->value does not work on a vector whose components are non-constant formulas.)

The operators . and x are defined to be the dot product and cross product, respectively, if the operands are vectors.

The norm function computes the length of a vector, and the unit function computes the unit vector in the same direction as a vector.

The dot and cross products as well as all Vector functions (including norm and unit) are undefined so that students cannot enter them in answers.

BEGIN_PGML_SOLUTION
Solution explanation goes here.
END_PGML_SOLUTION

ENDDOCUMENT();

Solution

A solution should be provided here.