Vector Quadruple Product

Author

Eric W. Weisstein
December 3, 2004

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

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

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

Initialization

<<Utilities`Typesetting`

stp[a_,b_,c_]:=Dot[a,Cross[b,c]]

Setup

𝒜=Array[a,3];
ℬ=Array[b,3];
𝒞=Array[c,3];
𝒟=Array[d,3];

Lagrange's identity

Cross[𝒜,ℬ].Cross[𝒞,𝒟]-(𝒜.𝒞 ℬ.𝒟-𝒜.𝒟 ℬ.𝒞)//Simplify

Others

(𝒜×ℬ)·(𝒞×𝒟)==(𝒜·𝒞)(ℬ·𝒟)-(𝒜·𝒟)(ℬ·𝒞)

Subtract@@{(𝒜\[Cross]ℬ).(𝒞\[Cross]𝒟),(𝒜.𝒞)(ℬ.𝒟)-(𝒜.𝒟)(ℬ.𝒞)}//Simplify

(𝒜×ℬ)·(𝒜×ℬ)==(𝒜·𝒜)(ℬ·ℬ)-(((𝒜·ℬ))^2)

Subtract@@{𝒜\[Cross]ℬ.𝒜\[Cross]ℬ,𝒜.𝒜 ℬ.ℬ-(((𝒜 .ℬ))^2)}//Simplify

𝒜×(ℬ×(𝒞×𝒟))==ℬ(𝒜·(𝒞×𝒟))-(𝒜·ℬ)(𝒞×𝒟)

Subtract@@{𝒜\[Cross](ℬ\[Cross](𝒞\[Cross]𝒟)),ℬ(𝒜.(𝒞\[Cross]𝒟))-(𝒜.ℬ)(𝒞\[Cross]𝒟)}//Simplify

(𝒜×ℬ)×(𝒞×𝒟)==(𝒞×𝒟)×(ℬ×𝒜)

Subtract@@{(𝒜\[Cross]ℬ)\[Cross](𝒞\[Cross]𝒟),(𝒞\[Cross]𝒟)\[Cross](ℬ\[Cross]𝒜)}//Simplify

Subtract@@{(𝒜\[Cross]ℬ)\[Cross](𝒞\[Cross]𝒟),stp[𝒜,ℬ,𝒟]𝒞-stp[𝒜,ℬ,𝒞]𝒟}//Simplify

Subtract@@{(𝒜\[Cross]ℬ)\[Cross](𝒞\[Cross]𝒟),stp[𝒞,𝒟,𝒜]ℬ-stp[𝒞,𝒟,ℬ]𝒜}//Simplify