Let $n\geq d$ and consider the Set
$$S_{np}=\{A \in \mathbb{R}^{n \times p}\lvert A^TA=I_d \}.$$
Does the function $d\colon S_{np} \times S_{np} \rightarrow \mathbb{R}$ defined by
$$d(A,B)=\sqrt{1-\det(A^TB)}$$
define a pseudometric on $S_{np}$? A pseudometric satisfies all conditions of a metric except that two elements can also have distance zero. Furthermore assume the equivalence relation $A \sim B$ if there exist an orthogonal $Q \in \mathbb{R}^{d \times d}$ with $A=BQ$. The set $S_{np}$ together with the equivalence relation can be identified with the grassmannian manifold. Does $d$ define a metric on the grassmanian manifold? This question interests me because im trying to approximate (interpolate) functions which take values in the grassmanian manifold and the above metric would open up a possibility for approximating such functions.
The difficult part is the triangle-inequality, i.e. for all $A,B,C \in S_{np}$ we need to prove that
$$\sqrt{1-\det(A^TC)}\leq \sqrt{1-\det(A^TB)}+\sqrt{1-\det(B^TC)}$$

Thanks for any help in advance.