vector triple product a b c vector

The volume of a parallelepiped with sidesA,BandCis the area of its base (say the parallelogram with area |BcrossC| ) multiplied by its altitude, the component ofAin the direction ofBcrossC.This is the magnitude ofAdotBcrossC;but it is also the magnitude of the determinant of the matrix with columnsA,BandC, so these linear functions of the vectors here are the same up to sign.
Adot(BcrossC) = det(A, B, C)

Thisvector tripple Productis not changed by cyclically permuting the vectors (for example toB,C,A) or by reversing the order of the factors in the dot product.

We can deduce then thatAdotBcrossC = CdotAcrosB =AcrossBdotC.In words,we can switch the dot and cross product without changing anything in this entity.(In either formula of course you must take the cross product first.) This product, like the determinant, changes sign if you just reverse the vectors in thecross product.

Thevector triple product,Across(BcrossC) is a vector, is normal toAand normal toBcrossCwhich means it is in the plane ofBandC. And it is linear in all three vectors.