## DBsubject('Calculus')
## DBchapter('Vector Geometry')
## DBsection('Dot Product and the Angle Between Two Vectors')
## KEYWORDS('calculus', 'parametric', 'vector', 'dot product', 'scalar product', 'angle', 'projection', 'proj')
## TitleText1('Calculus: Early Transcendentals')
## EditionText1('2')
## AuthorText1('Rogawski')
## Section1('12.3')
## Problem1('13')
## Author('Christopher Sira')
## Institution('W.H.Freeman')

DOCUMENT();
loadMacros("PG.pl","PGbasicmacros.pl","PGanswermacros.pl");
loadMacros("PGchoicemacros.pl");
loadMacros("Parser.pl");
loadMacros("freemanMacros.pl");
$context = Context("Vector");
Context()->texStrings;

sub vdot
{
    my($temp) = 0;

    for ($i = 0; $i < 3; $i++)
    {
        $temp = $temp + $_[$i] * $_[$i + 3];
    }
    return $temp
}

sub vlen
{
    return sqrt($_[0]**2 + $_[1]**2 + $_[2]**2);
}

sub vcos
{
    my($dot) = vdot(@_);

    return $dot / (vlen($_[0], $_[1], $_[2]) * vlen($_[3], $_[4], $_[5]));
}

sub vang
{
    return Formula("arccos(x)")->eval(x=>vcos(@_));
}

# projection of u onto v (first onto second)
sub vproj
{
    my(@temp) = ($_[3], $_[4], $_[5]);
    my($temp2) = vdot(@_);

    $temp2 = $temp2 / vdot($_[3], $_[4], $_[5], $_[3], $_[4], $_[5]);

    $temp[0] *= $temp2;
    $temp[1] *= $temp2;
    $temp[2] *= $temp2;

    return @temp;
}

# perpendicular component of u onto v (first onto second)
sub vperp
{
    my(@temp) = vproj(@_);

    $temp[0] = -1 * $temp[0] + $_[0];
    $temp[1] = -1 * $temp[1] + $_[1];
    $temp[2] = -1 * $temp[2] + $_[2];

    return @temp;
}

$iv0 = 1;
$iv1 = 0;
$iv2 = 0;

$jv0 = 0;
$jv1 = 1;
$jv2 = 0;

$kv0 = 0;
$kv1 = 0;
$kv2 = 1;

@iva = ($iv0, $iv1, $iv2);
@jva = ($jv0, $jv1, $jv2);
@kva = ($kv0, $kv1, $kv2);

$iv = Vector(@iva);
$jv = Vector(@jva);
$kv = Vector(@kva);

$v0 = Real(random(-10, 10, 1));
$v1 = Real(random(-10, 10, 1));
$v2 = Real(random(-10, 10, 1));

# make unit vector
#$v0 = $v0 / vlen($v0, $v1, $v2);
#$v1 = $v1 / vlen($v0, $v1, $v2);
#$v2 = $v2 / vlen($v0, $v1, $v2);


$w0 = Real(random(1, 10, 1));
$w1 = Real(random(1, 10, 1));
$w2 = Real(random(1, 10, 1));

# make unit vector
#$w0 = $v0 / vlen($v0, $v1, $v2);
#$w1 = $v1 / vlen($v0, $v1, $v2);
#$w2 = $v2 / vlen($v0, $v1, $v2);

@va = ($v0, $v1, $v2);
@wa = ($w0, $w1, $w2);

$v = Vector($v0, $v1, $v2);
$w = Vector($w0, $w1, $w2);

$mc = new_multiple_choice();

$ans = "obtuse";
$temp = "<";

$dot = vdot(@va, @wa);

if ($dot == 0)
{
    $mc->qa("The angle between \( $v \) and \( $w \) is:",
            "orthogonal");
    $mc->extra("acute", "obtuse");
    
    $ans = "orthogonal";
    $temp = "=";

}
else
{
    if ($dot < 0)
    {
        $mc->qa("The angle between \( $v \) and \( $w \) is:",
                "obtuse");
        $mc->extra("acute", "orthogonal");
        $ans = "obtuse";
        $temp = "<";
 
    }
    else
    {
        $mc->qa("The angle between \( $v \) and \( $w \) is:",
                "acute");
        $mc->extra("obtuse", "orthogonal");
        $ans = "acute";
        $temp = ">";
    }
}

Context()->texStrings;
BEGIN_TEXT
\{ beginproblem() \}
\{ textbook_ref_exact("Rogawski ET 2e", "12.3","13") \}
\{$mc->print_q\}
$PAR
\{$mc->print_a\}
END_TEXT
Context()->normalStrings;

ANS(str_cmp($mc->correct_ans));

Context()->texStrings;
SOLUTION(EV3(<<'END_SOLUTION'));
$PAR
$SOL
The type of angle between \( v \) and \( w \) can be determined by taking the dot product.
$PAR
\( v \cdotp w = 0 \) only when \( v \) and \( w \) are orthogonal.
$PAR
\( v \cdotp w < 0 \) only when the angle between \( v \) and \( w \) is obtuse.
$PAR
\( v \cdotp w > 0 \) only when the angle between \( v \) and \( w \) is acute.
$PAR
Now, \( $v \cdotp $w = $dot $temp 0 \), so the angle between \( v \) and \( w \) is ${ans}.
END_SOLUTION

ENDDOCUMENT();