Cross Product of Vectors
Cross Product of Vectors
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.
w=u×v
u
v
u·v
u×v
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.
w=u×v
|w|=|u||v|sinθ
θ
u
v
u
v
w
θ ≠ 0
u
v
u
v
w
Algebraically: In Cartesian coordinates, the components of the cross product can be read off a determinant, , where , , are the Cartesian unit vectors and , .
3×3
w=
i | j | k |
a | b | c |
d | e | f |
i=(1,0,0)
j=(0,1,0)
k=(0,0,1)
u=(a,b,c)
v=(d,e,f)