(1) We know that
X ( X ) = ( .) – (. )
It is also called as Lagrange's formula.
What is ( X ) X ?
So, ( X ) X | = | – X(X) |
= | –(.) + (.) (applying Lagrange's formula) |
(2) Prove X( X ) + X( X ) + X( X ) = 0
(It is known as Jacobi identity for cross product).
Applying Lagrange's formula and knowing that dot product is commutative, we have
LHS | = | X( X ) + X( X ) + X( X ) |
= | (.) – (.) + (.) – (.) + (.) – (.) | |
= | (.) – (.) + (.) – (.) + (.) – (.) | |
= | 0 | |
= | RHS |
(3) X )X = X ( X ) – X( X )
Equations at (2) and (3) are useful in vector calculation in Physics
(4) In Vector calculus and Faraday's laws in physics. There is a term called gradient.
It is denoted by the nabla symbol ∇ and pronounced 'del'.
Gradient is defined for functions of multiple variables,
while derivative is defined on functions of single variable.
So we have∇.(∇ X f) = ∇(∇.f) – (∇.∇)f
(5) In tensor calculus (which is an extention of vector calculus), the triple product using Levi-Civita symbol is
.( X ) = εijkaibjck