A third kind of ``products'' between two Euclidean vectors $\vec{a}$ and $\vec{b}$ , besides the scalar product $\vec{a}\!\cdot\!\vec{b}$ and the vector product $\vec{a}\!\times\!\vec{b}$ , is the dyad product $\vec{a}\,\vec{b}$ , which is usually denoted without any multiplication symbol. The dyad products and the finite formal sums

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A dyad is not a vector, but an operator. It functions on any vector $\vec{v}$ producing from it new vectors or new dyads according to the definitions

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The dyad product obeys the distributive laws $$\vec{a}(\vec{b}\!+\!\vec{c}) = \vec{a}\,\vec{b}\!+\!\vec{a}\,\vec{c}, \quad (\vec{b}\!+\!\vec{c})\vec{a} = \vec{b}\,\vec{a}\!+\!\vec{c}\,\vec{a},$$ which can be verified by multiplying an arbitrary vector $\vec{v}$ and both sides of these equations and then comparing the results. Likewise, the scalar factor transfer rule is valid. It follows that if we have $\vec{a} = a_1\vec{e_1}+a_2\vec{e_2}+a_3\vec{e_3}$ and $\vec{b} = b_1\vec{e_1}+b_2\vec{e_2}+b_3\vec{e_3}$ in the orthonormal basis $\{\vec{e_1},\,\vec{e_2},\,\vec{e_3}\}$ (for the brevity, we confine us to vectors of $\mathbb{R}^3$ ), their dyad product is the sum \begin{eqnarray*} \vec{a}\,\vec{b} = &a_1b_1\vec{e_1}\vec{e_1}+a_1b_2\vec{e_1}\vec{e_2}+a_1b_3\vec{e_1}\vec{e_3}+\\ &a_2b_1\vec{e_2}\vec{e_1}+a_2b_2\vec{e_2}\vec{e_2}+a_2b_3\vec{e_2}\vec{e_3}+\\ &a_3b_1\vec{e_3}\vec{e_1}+a_3b_2\vec{e_3}\vec{e_2}+a_3b_3\vec{e_3}\vec{e_3},\;\; \end{eqnarray*}which shows that the dyad product has been formed similarly as the matrix product of the vectors $(a_1,\,a_2,\,a_3)^{\mbox{\scriptsize{T}}}$ and $(b_1,\,b_2,\,b_3)$ .

The unit dyad $$\mbox{I} := \vec{e_1}\vec{e_1}\!+\!\vec{e_2}\vec{e_2}\!+\!\vec{e_3}\vec{e_3} = \nabla\vec{r},$$ where $\vec{r}$ is the position vector, satisfies $$\mbox{I}\!\cdot\!\vec{v} = \vec{v}\!\cdot\!\mbox{I} = \vec{v}$$ and $$\mbox{I}\!\times\!(\vec{u}\!\times\!\vec{v}) = \vec{v}\,\vec{u}-\vec{u}\,\vec{v}$$ for all vectors $\vec{u}$ and $\vec{v}$ .

The product of two dyad products $\vec{a}\,\vec{b}$ and $\vec{c}\,\vec{d}$ is defined to be the dyad

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Bibliography

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K. V¨AISÄLÄ: Vektorianalyysi. Werner Söderström Osakeyhtiö, Helsinki (1961).