Scalar Triple Product

Author

Eric W. Weisstein
April 14, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Algebra/ScalarTripleProduct.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/ScalarTripleProduct.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Examples

𝒜=Array[a,3];
ℬ=Array[b,3];
𝒞=Array[c,3];
STP[𝒜_,ℬ_,𝒞_]:=𝒜.Cross[ℬ,𝒞]

Subtract@@{STP[𝒜,ℬ,𝒞],ℬ.Cross[𝒞,𝒜]}//Simplify

Subtract@@{STP[𝒜,ℬ,𝒞],𝒞.Cross[𝒜,ℬ]}//Simplify

Transpose@{𝒜,ℬ,𝒞}//MatrixForm

Subtract@@{STP[𝒜,ℬ,𝒞],Det[Transpose@{𝒜,ℬ,𝒞}]}//Simplify

{𝒜,ℬ,𝒞}//MatrixForm

Subtract@@{STP[𝒜,ℬ,𝒞],Det[{𝒜,ℬ,𝒞}]}//Simplify

Subtract@@{STP[𝒜,ℬ,𝒞],Det[{𝒜,ℬ,𝒞}]}//Simplify

<<DiscreteMath`Combinatorica`

Permutations[Range[3]]

Plus@@Function[{i,j,k},Signature[{i,j,k}]a[i]b[j]c[k]]@@@Permutations[Range[3]]

Subtract@@{STP[𝒜,ℬ,𝒞],Plus@@Function[{i,j,k},Signature[{i,j,k}]a[i]b[j]c[k]]@@@Permutations[Range[3]]}//Simplify