Final answer:
To calculate A(AB) AC for observables, the correct application of the commutative and distributive laws, especially the anticommutative property of the cross product, is essential. The order in which the cross product is taken matters, resulting in vector terms grouped by common unit vectors and factoring to aid in calculation.
Step-by-step explanation:
When calculating A(AB) AC for given observables, it is crucial to consider the algebraic properties of the operations involved. The commutative and distributive laws need to be applied accurately in the context of vector multiplication. Specifically, for the cross product, which is anticommutative, the order of multiplication is essential, unlike the dot product which is commutative.
To find the cross product, one can use the following formula:
Č = Ả × B = (Ay Bz − Az By)Î + (Az Bx − Ax Bz)ǂ + (Ax By − AyBx)Ê.
In addition, when observables are involved, we must be mindful of the correct symbolic representation, ensuring that we recognize when expressions are vectors versus scalars, such as in the commutation relations provided initially.