cross product

cross product

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imaginary part

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real part

\PMlinkescapephrase

spanned by

The

cross product

(or

vector product

) of two vectors in

ℝ

3

is a vector

http://planetmath.org/node/1285

orthogonal

to the plane of the two vectors being crossed, whose magnitude is equal to the area of the

parallelogram

defined by the two vectors. Notice there can be two such vectors. The cross product produces the vector that would be in a

right-handed coordinate system

with the plane.

We write the cross product of the vectors

𝐚

and

𝐛

as

𝐚

×

𝐛

=

det

⁡

(

𝐢

𝐣

𝐤

a

1

a

2

a

3

b

1

b

2

b

3

)

=

(

a

2

⁢

b

3

-

a

3

⁢

b

2

)

⁢

𝐢

+

(

a

3

⁢

b

1

-

a

1

⁢

b

3

)

⁢

𝐣

+

(

a

1

⁢

b

2

-

a

2

⁢

b

1

)

⁢

𝐤

with

𝐚

=

a

1

⁢

𝐢

+

a

2

⁢

𝐣

+

a

3

⁢

𝐤

and

𝐛

=

b

1

⁢

𝐢

+

b

2

⁢

𝐣

+

b

3

⁢

𝐤

, where

(

𝐢

,

𝐣

,

𝐤

)

is a

right-handed

orthonormal basis

for

ℝ

3

.

If we regard vectors in

ℝ

3

as quaternions with real part equal to zero, with

i

=

𝐢

,

j

=

𝐣

and

k

=

𝐤

, then the cross product of two vectors can be obtained by zeroing the real part of the product of the two quaternions. (A

similar

construction using octonions instead of quaternions gives a “cross product” in

ℝ

7

which shares many of the properties of the

ℝ

3

cross product.)

If we write vectors in the form

𝐚

=

(

a

1

a

2

a

3

)

, then we can express the cross product as

𝐚

×

𝐛

=

(

0

-

a

3

a

2

a

3

0

-

a

1

-

a

2

a

1

0

)

⁢

𝐛

.

The

spectrum

of this matrix
(and therefore of the map

𝐛

↦

𝐚

×

𝐛

)
is

{

0

,

i

⁢

|

𝐚

|

,

-

i

⁢

|

𝐚

|

}

.

Properties of the cross product

In the following,

𝐚

,

𝐛

and

𝐜

will be arbitrary vectors in

ℝ

3

, and

s

and

t

will be arbitrary

real numbers

.

•

𝐚

×

𝐚

=

0

.

•

𝐚

×

(

𝐛

×

𝐜

)

+

𝐛

×

(

𝐜

×

𝐚

)

+

𝐜

×

(

𝐚

×

𝐛

)

=

0

.

•

The cross product is a bilinear map.
This means that

(

s

⁢

𝐚

)

×

(

t

⁢

𝐛

)

=

(

s

⁢

t

)

⁢

(

𝐚

×

𝐛

)

,
and that the cross product is

distributive

over

vector addition

,
that is,

𝐚

×

(

𝐛

+

𝐜

)

=

𝐚

×

𝐛

+

𝐚

×

𝐜

and

(

𝐛

+

𝐜

)

×

𝐚

=

𝐛

×

𝐚

+

𝐜

×

𝐚

.

•

The three properties above mean that the cross product makes

ℝ

3

into a Lie algebra.

•

𝐚

×

𝐛

is orthogonal to both

𝐚

and

𝐛

.

•

𝐚

×

𝐛

=

-

𝐛

×

𝐚

.

•

The length of

𝐚

×

𝐛

is the area of the parallelogram spanned by

𝐚

and

𝐛

, so

|

𝐚

×

𝐛

|

=

|

𝐚

|

⁢

|

𝐛

|

⁢

sin

⁡

θ

, where

θ

is the angle between

𝐚

and

𝐛

. This gives us an expression for the area of a

triangle

in

ℝ

3

: if the vertices are at

𝐚

,

𝐛

and

𝐜

, then the area is

1

2

⁢

|

(

𝐚

-

𝐜

)

×

(

𝐛

-

𝐜

)

|

, which can be written more symmetrically as

1

2

⁢

|

𝐚

×

𝐛

+

𝐛

×

𝐜

+

𝐜

×

𝐚

|

.

•

From the above, you can see that the cross product of any vector with

𝟎

is

𝟎

. More generally, the cross product of two

parallel vectors

is

𝟎

, since

sin

⁡

0

=

0

.

•

One can also see that

|

𝐚

×

𝐛

|

2

=

|

𝐚

|

2

⁢

|

𝐛

|

2

-

|

𝐚

⋅

𝐛

|

2

.

•

𝐚

×

(

𝐛

×

𝐜

)

=

(

𝐚

⋅

𝐜

)

⁢

𝐛

-

(

𝐚

⋅

𝐛

)

⁢

𝐜

.
This is the

vector triple product

.

•

The cross product is rotationally

invariant

(

http://planetmath.org/RotationalInvarianceOfCrossProduct

).
That is, for any

3

×

3

rotation matrix

M

we have

M

⁢

(

𝐚

×

𝐛

)

=

(

M

⁢

𝐚

)

×

(

M

⁢

𝐛

)

.

Title

cross product

Canonical name

CrossProduct

Date of creation

2013-03-22 11:58:52

Last modified on

2013-03-22 11:58:52

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

46

Author

yark (2760)

Entry type

Definition

Classification

msc 15A90

Classification

msc 15A72

Synonym

vector product

Synonym

outer product

Related topic

Vector

Related topic

DotProduct

Related topic

TripleScalarProduct

Related topic

ExteriorAlgebra

Related topic

DyadProduct

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