Product triple vector

Triple products involving vectors arise often in physical problems. One such product is (A x B) x C, which is clearly represented by a vector. It is therefore called the vector triple product, whose development can be made as follows. If, in a Cartesian system, the vector A is chosen to be coMnear with the x direction, A = Axi. The vector B can, without loss of generality, be placed in the x,y plane. It is then given by B = Bxi + Byj. The vector C is then in a general direction, as given by C = Cxi + Cyj + Czk, as shown in Fig. 6. Then, the cross products can be easily developed in the form A x B — AxByk and... 

The product Vi x (V2 x V3) defines the vector triple product. Obviously, in this case, the brackets are vital to the definition. 

The derivation of the quantum mechanical expression for the rotational g tensor requires the derivation of quantmn mechanical expressions for the rigid and induced contribution to the rotational magnetic moment. An expression for the first is most easily derived in analogy to the electric dipole moment in Section 4.3 by translating the classical expression, Eq. (6.2), to quantum mechanics. Before doing so, however, we want to make use of Lagrange s formula for a vector triple product [see Exercise 6.11... 

By this means we can reduce any scalar-valued expression to a polynomial in the inner products of single pairs of vectors, together with vector triple products, that is only linear in the triple products. Thus any set of multivectors can be characterized, up to rotation, by specifying the scalar values of a system of such multivariate polynomials. 

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