Dot Product between 2 Dyads/Dyadics

1. Calculating Dot Product between 2 Dyads/Dyadics is same as Mutiplying the 2 Dyad/Dyadic Matrices. For calculating Dot Product between 2 Dyads/Dyadics the Order of the both Dyads/Dyadics must be same. The result of Dot Product between 2 Dyads/Dyadics is another Dyad/Dyadic.
2. Given a Dyad \(\overleftrightarrow{AB}\) formed of Vectors \(\vec{A}\) and \(\vec{B}\) and another Dyad \(\overleftrightarrow{CD}\) formed of Vectors \(\vec{C}\) and \(\vec{D}\), the following relations hold true
   
   \(\overleftrightarrow{AB} \cdot \overleftrightarrow{CD}= [\overleftrightarrow{AB}] [\overleftrightarrow{CD}]=(\vec{B} \cdot \vec{C})(\vec{A} \otimes_{T} \vec{D})\)
   
   \(\overleftrightarrow{CD} \cdot \overleftrightarrow{AB}= [\overleftrightarrow{CD}] [\overleftrightarrow{AB}]=(\vec{D} \cdot \vec{A})(\vec{C} \otimes_{T} \vec{B})\)
3. Given a Dyadic Polynomial Matrix \(\overleftrightarrow{AB}\) formed by addition of Dyads \(\overleftrightarrow{A_1B_1}\), \(\overleftrightarrow{A_2B_2}\), ... , \(\overleftrightarrow{A_KB_K}\) and another Dyadic Polynomial Matrix \(\overleftrightarrow{CD}\) formed by addition of Dyads \(\overleftrightarrow{C_1D_1}\), \(\overleftrightarrow{C_2D_2}\), ... , \(\overleftrightarrow{C_ND_N}\), the following relations hold true
   
   \(\overleftrightarrow{AB} \cdot \overleftrightarrow{CD}= [\overleftrightarrow{AB}] [\overleftrightarrow{CD}]=\sum_{i=1}^{K}\sum_{j=1}^{N}(\vec{B_i} \cdot \vec{C_j})(\vec{A_i} \otimes_{T} \vec{D_j})\)
   
   \(\overleftrightarrow{CD} \cdot \overleftrightarrow{AB}= [\overleftrightarrow{CD}] [\overleftrightarrow{AB}]=\sum_{i=1}^{K}\sum_{j=1}^{N}(\vec{D_j} \cdot \vec{A_i})(\vec{C_j} \otimes_{T} \vec{B_i})\)
4. The Dot Product between 2 Dyads/Dyadics is Non-Commutative, that is for any 2 Non Identity Dyad/Dyadic \(\overleftrightarrow{AB}\) and \(\overleftrightarrow{CD}\)
   
   \(\overleftrightarrow{AB} \cdot \overleftrightarrow{CD} \neq \overleftrightarrow{CD} \cdot \overleftrightarrow{AB}\)
5. Given 2 3-Dimensional Vectors \(\vec{A}\) and \(\vec{B}\) , the following relations related to Identity Dyadic hold true
   
   \((\overleftrightarrow{I} \times \vec{A}) \cdot (\overleftrightarrow{I} \times \vec{B})= (\vec{A} \times \overleftrightarrow{I}) \cdot (\vec{B} \times \overleftrightarrow{I} )=[\vec{A}_{\times}][\vec{B}_{\times}] = \vec{B} \otimes_{T} \vec{A} - (\vec{A} \cdot \vec{B}) \overleftrightarrow{I}\)
   
   \((\overleftrightarrow{I} \times \vec{B}) \cdot (\overleftrightarrow{I} \times \vec{A})= (\vec{B} \times \overleftrightarrow{I}) \cdot (\vec{A} \times \overleftrightarrow{I} )=[\vec{B}_{\times}][\vec{A}_{\times}] = \vec{A} \otimes_{T} \vec{B} - (\vec{A} \cdot \vec{B}) \overleftrightarrow{I}\)
   
   In above equations \(\vec{A}_{\times}\) and \(\vec{B}_{\times}\) are Skew-Symmetric Matrices corresponding to Vectors \(\vec{A}\) and \(\vec{B}\) respectively.