Manuals/calci/VECTORDIRECTPRODUCT

VECTORDIRECTPRODUCT (a,b)

• $a$ and $b$ any two set of values.

Description

• This function shows the Vector Direct product.
• The third type of vector multiplication is called the direct product, and is written AB.
• In $VECTORDIRECTPRODUCT(a,b)$, $a$ and $b$ are the two vectors.
• Multiplying one vector by another under the direct product gives a tensor result.
• The rectangular components of the direct product may be found by matrix multiplication: one multiplies the column vector A by the transpose of B, which gives a 3X3 matrix:

$AB=AB^{T}$= ${\begin{pmatrix}A_{x}\\A_{y}\\A_{z}\end{pmatrix}}$$(B_{x}B_{y}B_{Z})$=${\begin{pmatrix}A_{x}B_{x}&A_{x}B_{y}&A_{x}B_{z}\\A_{y}B_{x}&A_{y}B_{y}&A_{y}B_{z}\\A_{z}B_{x}&A_{z}B_{y}&A_{z}B_{z}\end{pmatrix}}$

• The direct product is non-commutative $(AB\neq BA)$.
• A few vector product identities are of interest:

1. $A.BXC=AXB.C=B.CXA=BXC.A=C.AXB$
2. $AX(BXC)=B.(A.C)-C(A.B)$
3. $(AXB)XC=B(A.C)-A(B.C)$
4. $(AXB).(CXD)=(A.C)(B.D)-(A.D)(B.C)$
5. $(AXB).(CXD)=(AxB.D)C-(AxB.C)D$