dyad product

A third kind of “products” between two Euclidean vectors a and b, besides the scalar productMathworldPlanetmath ab and the vector product a×b, is the  dyad productab,  which is usually denoted without any multiplication symbol.  The dyad products and the finite formal sums

Φ:=μaμbμ (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:=μaμ(bμ*v),v*Φ:=μ(v*aμ)bμ. (2)

Here the asterisks empty, in which case the vector v must be replaced by a scalar v; the products Φv and vΦ are dyads.

The dyad product obeys the distributive laws


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=a1e1+a2e2+a3e3  and  b=b1e1+b2e2+b3e3  in the orthonormal basis{e1,e2,e3} (for the brevity, we confine us to vectors of 3), their dyad product is the sum

ab= a1b1e1e1+a1b2e1e2+a1b3e1e3+

which shows that the dyad product has been formed similarly as the matrix productMathworldPlanetmath of the vectors  (a1,a2,a3)T  and (b1,b2,b3).

The unit dyad


where r is the position vector,




for all vectors u and v.

The product of two dyad products  ab  and  cd  is defined to be the dyad

(ab)(cd):=(bc)(ad) (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:



  • 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