A.(A*B) is the box product of [A A B] expanding it into its matrix form we will get 2 identical rows hence A.(A*B) is zero.
also geometrically if 3 vectors lie in the same plane their box product is zero . A,A, B lie in the same plane (A and B lie in the same plane Because any 2 vectors lie in the same plane) so A.(A*B) is zero
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