A third kind of “products” between two Euclidean vectors and , besides the scalar product and the vector product , is the dyad product , which is usually denoted without any multiplication symbol. The dyad products and the finite formal sums
of them are called dyads.
A dyad is not a vector, but an operator. It on any vector producing from it new vectors or new dyads according to the definitions
Here the asterisks empty, in which case the vector must be replaced by a scalar ; the products and are dyads.
The dyad product obeys the distributive laws
which can be verified by multiplying an arbitrary vector and both of these equations and then comparing the results. Likewise, the scalar factor transfer rule is valid. It follows that if we have and in the orthonormal basis (for the brevity, we confine us to vectors of ), their dyad product is the sum
which shows that the dyad product has been formed similarly as the matrix product of the vectors and .
The unit dyad
where is the position vector,
for all vectors and .
The product of two dyad products and is defined to be the dyad
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).
|Date of creation||2013-03-22 15:26:44|
|Last modified on||2013-03-22 15:26:44|
|Last modified by||pahio (2872)|
|Defines||product of dyads|