This Demonstration computes and displays the cross product
(black) of two vectors
(red) and
(blue) in three dimensions. The dot product
of the vectors is a scalar (number), while the cross product
is a vector.
The cross product can be defined in several equivalent ways.
Geometrically: (1) The length of the vector
is given by
, where
is the angle between
and
. (The length is equal to the area of the parallelogram spanned by the vectors
and
.) (2) The direction of
, when
, is perpendicular to both
and
, oriented in the sense that
,
,
form a right-handed system.
Algebraically: In Cartesian coordinates, the components of the cross product can be read off a
determinant,
, where
,
,
are the Cartesian unit vectors and
,
.
