(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