the area between 2 vectors is | a x b |.
we could square the area on the following fomura.
( a dot b ) = | a | | b | cos( theta )
( 1, 0, 0 ) x ( 0, 0, 1 ) = ( 0, 1, 0 ) means at least ( ?, x1*z2, ? ).
( 0, 0, 1 ) x ( 1, 0, 0 ) = ( 0, -1, 0 ) means at least ( ?, -z1*x2, ? ).
( 0, 1, 0 ) x ( 1, 0, 0 ) = ( 0, 0, -1 ) means at least ( ?, ?, y1*x2 ).
( 1, 0, 0 ) x ( 0, 1, 0 ) = ( 0, 0, 1 ) means at least ( ?, ?, -x1*y2 ).
( 0, 0, 1 ) x ( 0, 1, 0 ) = ( 1, 0, 0 ) means at least ( z1*y2, ?, ? ).
( 1, 0, 0 ) x ( 0, 0, 1 ) = ( -1, 0, 0 ) means at least ( -x1*z2, ?, ? ).
a and b are not unit vectors.
cross( a/|a|, b/|b| )x = { ( z1/|a| ) * ( y2/|b| ) - ( y1/|a| ) * ( z2/|b| ) }
cross( a/|a|, b/|b| )y = { ( x1/|a| ) * ( z2/|b| ) - ( z1/|a| ) * ( x2/|b| ) }
cross( a/|a|, b/|b| )z = { ( y1/|a| ) * ( x2/|b| ) - ( x1/|a| ) * ( y2/|b| ) }
, which concludes the below.
cross( a, b )x = { ( z1 ) * ( y2 ) - ( y1 ) * ( z2 ) }
cross( a, b )y = { ( x1 ) * ( z2 ) - ( z1 ) * ( x2 ) }
cross( a, b )z = { ( y1 ) * ( x2 ) - ( x1 ) * ( y2 ) }
( 1, 1, 0 ) x ( -1, 1, 0 ) = ( 0, 0, -2 ) on the above.
