We introduce three concepts, called $I$-vector, $IP$-vector, and $P$-vector, which are related to isosceles orthogonality and Pythagorean orthogonality in normed linear spaces. Having the Banach-Mazur rotation problem in mind, we prove that an almost transitive real Banach space, whose dimension is at least three and which contains an $I$-vector (an $IP$-vector, a $P$-vector, or a unit vector whose pointwise James constant is $\sqrt2$, respectively) is a Hilbert space.