Cross-Dot Product between 2 Dyads/Dyadics

1. For calculating Cross-Dot Product between 2 Dyads/Dyadics the Order of the both Dyads/Dyadics must be 3. The result of Cross-Dot Product between 2 Dyads/Dyadics is a Vector.
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
   
   \(\overleftrightarrow{AB} {}_{\,\centerdot}^\times \overleftrightarrow{CD}= (\vec{A} \times \overleftrightarrow{CD}) \cdot \vec{B}= (\vec{A} \times \vec{C})(\vec{B} \cdot \vec{D})\)
   
   \(\overleftrightarrow{CD} {}_{\,\centerdot}^\times \overleftrightarrow{AB}= (\vec{C} \times \overleftrightarrow{AB}) \cdot \vec{D}= (\vec{C} \times \vec{A})(\vec{D} \cdot \vec{B})\)
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
   
   \(\overleftrightarrow{AB} {}_{\,\centerdot}^\times \overleftrightarrow{CD}= \sum_{i=1}^{K}\sum_{j=1}^{N}(\vec{A_i} \times \overleftrightarrow{C_jD_j}) \cdot \vec{B_i}= \sum_{i=1}^{K}\sum_{j=1}^{N}(\vec{A_i} \times \vec{C_j})(\vec{B_i} \cdot \vec{D_j})\)
   
   \(\overleftrightarrow{CD} {}_{\,\centerdot}^\times \overleftrightarrow{AB}= \sum_{i=1}^{K}\sum_{j=1}^{N}(\vec{C_j} \times \overleftrightarrow{A_iB_i}) \cdot \vec{D_j}= \sum_{i=1}^{K}\sum_{j=1}^{N}(\vec{C_j} \times \vec{A_i})(\vec{D_j} \cdot \vec{B_i})\)
4. The Cross-Dot Product between 2 Dyads/Dyadics is Anti-Commutative, that is for any 2 Dyad/Dyadic \(\overleftrightarrow{AB}\) and \(\overleftrightarrow{CD}\)
   
   \(\overleftrightarrow{AB} {}_{\,\centerdot}^\times \overleftrightarrow{CD} = -\overleftrightarrow{CD} {}_{\,\centerdot}^\times \overleftrightarrow{AB}\)
5. Given a Dyad \(\overleftrightarrow{AB}\) formed from Vectors \(\vec{A}\) and \(\vec{B}\) and Identity Dyadic \(\overleftrightarrow{I}\), following relations hold true
   
   \(\overleftrightarrow{I} {}_{\,\centerdot}^\times \overleftrightarrow{AB}= \vec{B} \times \vec{A}=-\overleftrightarrow{AB} {}_{\,\centerdot}^\times \overleftrightarrow{I}\)
   
   \(\overleftrightarrow{I} {}_{\,\centerdot}^\times \overleftrightarrow{BA}= \vec{A} \times \vec{B}=-\overleftrightarrow{BA} {}_{\,\centerdot}^\times \overleftrightarrow{I}\)