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 , .