Cross Product of Vectors

a

b

c

d

e

f

{1.53,0.22,0.65}



{-0.82,-1.88,-1.04}

=

{0.99,1.06,-2.7}

This Demonstration computes and displays the cross product

w=u×v

(black) of two vectors

u

(red) and

v

(blue) in three dimensions. The dot product

u·v

of the vectors is a scalar (number), while the cross product

u×v

is a vector.

The cross product can be defined in several equivalent ways.

Geometrically: (1) The length of the vector

w=u×v

is given by

|w|=|u||v|sinθ

, where

θ

is the angle between

u

and

v

. (The length is equal to the area of the parallelogram spanned by the vectors

u

and

v

.) (2) The direction of

w

, when

θ ≠ 0

, is perpendicular to both

u

and

v

, oriented in the sense that

u

,

v

,

w

form a right-handed system.

Algebraically: In Cartesian coordinates, the components of the cross product can be read off a

3×3

determinant,

w=

i	j	k
a	b	c
d	e	f



, where

i=(1,0,0)

,

j=(0,1,0)

,

k=(0,0,1)

are the Cartesian unit vectors and

u=(a,b,c)

,

v=(d,e,f)

.

Details

Snapshot 1: when

u

and

v

are in the

x

,

y

plane,

u×v

has only a

z

component

Snapshot 2: this shows the unit vector relationship

i×j=k

Snapshot 3: when

u

and

v

are collinear, their cross product vanishes

External Links

Dot Product

Cross Product (Wolfram MathWorld)

Right-Hand Rule (Wolfram MathWorld)

Permanent Citation

S. M. Blinder, Amy Blinder

"Cross Product of Vectors"
http://demonstrations.wolfram.com/CrossProductOfVectors/
Wolfram Demonstrations Project
Published: January 30, 2008