Manuals/calci/DYADIC

DYADIC(a,b)

OR VECTORDIRECTPRODUCT (a,b)

• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle VECTORDIRECTPRODUCT (a,b)} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle AB=AB^T} = Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{pmatrix} A_x \\ A_y \\ A_z \end{pmatrix}} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (AB \neq BA)} .
• A few vector product identities are of interest:

1. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A.BXC = AXB.C=B.CXA=BXC.A=C.AXB}
2. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle AX(BXC)= B.(A.C)- C(A.B)}
3. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (AXB)XC = B(A.C)-A(B.C)}
4. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (AXB).(CXD)= (A.C)(B.D)-(A.D)(B.C)}
5. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (AXB).(CXD) = (AxB.D)C-(AxB.C)D}

Examples

1. DYADIC([1,2,3],[8,7,6]) = 40
2. VECTORDIRECTPRODUCT([14,17,20],[22,26,5]) = 850
3. VECTORDIRECTPRODUCT([2.7,3.9,10.2],[14.5,19,-4]) = 72.45
4. DYADIC([-8,-4,2],[10,-45,67]) = 234

See Also

• DOTPRODUCT
• CROSSPRODUCT
• CARTESIANPRODUCT

References

| Direct Product