v1 * w1 + v2 * w2 + v3 * w3 + ... + vn * wn

##</code><code>##

  V * W
---------   = cos ( theta )
|V| * |W|

or

VV = V / |V|; WW = W / |W|

  VV * WW     VV * WW 
----------- = ------- = cos( theta )
|VV| * |WW|    1 * 1

##</code><code>##

$v = [ 1, 4, 7, 10, 13,  16, 19, 22 ];
$w = [ 1, 2, 4, 8, 16 ];

sub dot_prod {
    # report error unless args exist and are references to arrays
    return unless (  (2 = @_ )                   &&
                          (ref $_[0] eq "ARRAY") && 
                          (ref $_[1] eq "ARRAY") );

    # if either array is empty,  the dot product is zero.
    return 0 unless ( @{$_[0]} && @{$_[1]} );

    my @v = @{$_[0]};
    my @w = @{$_[1]};
    my $result = 0;
    my ( $v, $w ) = ( shift @v, shift @w );
    
    do { 
         if ( $v == $w ) {
            $v = shift @v;
            $w = shift @w;
            $result++;
         } elsif ( $v > $w ) {
             $w = shift @w;
         } elsif ( $v < $w ) {
	      $v = shift @v;
         }
    } while ( @v && @w );

    return $result;
}

sub norm {
   return 0 unless ( @_ && ( ref $_[0] eq "ARRAY ) )
   return sqrt( @{$_[0]} );
} 

my $dot_product = dot_prod( $v, $w ) or die( "dot_prod error" );;
my $denom = (norm( $v ) * norm( $w );
my $cos_theta = $denom ? $dot_product / $denom : 0;