Third equivalent form of the uncertainty principle is given by the scalar product of the uncertainty of angular position measured on the ���� plane and the uncertainty of the orthogonal component of the angular momentum along the ��-axis: ∆��·∆��ᶻ≥ℏ. The physical meaning as given by L. I. Schiff is that the precise measurement of the angular position of an electron in certain orbit around the nucleus carries with it the loss of all knowledge of the ��-component of its angular momentum (see L. I. Schiff, Quantum Mechanics, 2nd ed., Page 7, 1955). However, this uncertainty can be reduced simply by squaring both side of the inequality: (∆��·∆��ᶻ)²≥ℏ². Although the right hand side is now a much smaller physical constant, the physical meaning of the left hand side leaves much to be desired.

Mathematically, the scalar product of two vectors is equivalent to the product of their absolute values and the cosine of the angle between them: ∆��·∆��ᶻ=|∆�� ||·∆��ᶻ|������|��|. The square of this equivalent form is |∆�� |²|·∆��ᶻ|²������²|��| or using the trigonometric identity (������²��+������²��=1) then |∆�� |²|·∆��ᶻ|²������²|��|=|∆�� |²|·∆��ᶻ|² -|∆�� |²|·∆��ᶻ|²������²|��| for ��=0° or ���� where �� is always a positive integer then the square of the uncertainty is simply given by |∆�� |²|·∆��ᶻ|² which is equivalent to the product of two spacetime metric: (∆��²₁+∆��²₂)(|·∆��ᶻ|²) where the right factor is the square of angular momentum. It needed a true physical meaning in order to provide the determination of absolute certainty.