dyad product

dyad product

A third kind of “products” between two

Euclidean vectors

a

→

and

b

→

, besides the

scalar product

a

→

⋅

b

→

and the

vector product

a

→

×

b

→

, is the

dyad product

a

→

⁢

b

→

,  which is usually denoted without any

multiplication

symbol.  The dyad products and the finite formal sums

Φ

:=

∑

μ

a

→

μ

⁢

b

→

μ

(1)

of them are called

dyads

.

A dyad is not a vector, but an operator.  It on any vector

v

→

producing from it new vectors or new dyads according to the

definitions

Φ

*

v

→

:=

∑

μ

a

→

μ

⁢

(

b

→

μ

*

v

→

)

,

v

→

*

Φ

:=

∑

μ

(

v

→

*

a

→

μ

)

⁢

b

→

μ

.

(2)

Here the asterisks empty, in which case the vector

v

→

must be replaced by a scalar

v

; the products

Φ

⁢

v

and

v

⁢

Φ

are dyads.

The dyad product obeys the distributive laws

a

→

⁢

(

b

→

+

c

→

)

=

a

→

⁢

b

→

+

a

→

⁢

c

→

,

(

b

→

+

c

→

)

⁢

a

→

=

b

→

⁢

a

→

+

c

→

⁢

a

→

,

which can be verified by multiplying an arbitrary vector

v

→

and both of these equations and then comparing the results.  Likewise, the

scalar factor transfer rule

is valid.  It follows that if we have

a

→

=

a

1

⁢

e

1

→

+

a

2

⁢

e

2

→

+

a

3

⁢

e

3

→

and

b

→

=

b

1

⁢

e

1

→

+

b

2

⁢

e

2

→

+

b

3

⁢

e

3

→

in the

orthonormal basis

{

e

1

→

,

e

2

→

,

e

3

→

}

(for the brevity, we confine us to vectors of

ℝ

3

), their dyad product is the sum

a

→

⁢

b

→

=

a

1

⁢

b

1

⁢

e

1

→

⁢

e

1

→

+

a

1

⁢

b

2

⁢

e

1

→

⁢

e

2

→

+

a

1

⁢

b

3

⁢

e

1

→

⁢

e

3

→

+

a

2

⁢

b

1

⁢

e

2

→

⁢

e

1

→

+

a

2

⁢

b

2

⁢

e

2

→

⁢

e

2

→

+

a

2

⁢

b

3

⁢

e

2

→

⁢

e

3

→

+

a

3

⁢

b

1

⁢

e

3

→

⁢

e

1

→

+

a

3

⁢

b

2

⁢

e

3

→

⁢

e

2

→

+

a

3

⁢

b

3

⁢

e

3

→

⁢

e

3

→

,

which shows that the dyad product has been formed similarly as the

matrix product

of the vectors

(

a

1

,

a

2

,

a

3

)

T

and

(

b

1

,

b

2

,

b

3

)

.

The

unit dyad

I

:=

e

1

→

⁢

e

1

→

+

e

2

→

⁢

e

2

→

+

e

3

→

⁢

e

3

→

=

∇

⁡

r

→

,

where

r

→

is the

position vector

,

I

⋅

v

→

=

v

→

⋅

I

=

v

→

and

I

×

(

u

→

×

v

→

)

=

v

→

⁢

u

→

-

u

→

⁢

v

→

for all vectors

u

→

and

v

→

.

The

product

of two dyad products

a

→

⁢

b

→

and

c

→

⁢

d

→

is defined to be the dyad

(

a

→

⁢

b

→

)

⁢

(

c

→

⁢

d

→

)

:=

(

b

→

⋅

c

→

)

⁢

(

a

→

⁢

d

→

)

(3)

and the product of such dyads as (1) to be the formal sum of individual products (3).  The multiplication of dyads is associative and distributive over

addition

.  The unit dyad acts as unity in the ring of dyads:

I

⁢

Φ

=

Φ

⁢

I

=

Φ

∀

Φ

References

1

K. Väisälä:

Vektorianalyysi

.  Werner Söderström Osakeyhtiö, Helsinki (1961).

Title

dyad product

Canonical

name

DyadProduct

Date of creation

2013-03-22 15:26:44

Last modified on

2013-03-22 15:26:44

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

15

Author

pahio (2872)

Entry type

Definition

Classification

msc 15A72

Related topic

Frame

Related topic

DotProduct

Related topic

CrossProduct

Related topic

PositionVector

Related topic

KalleVaisala

Defines

dyad

Defines

unit dyad

Defines

product of dyads

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