  Given an orthogonal projection $P$ and a free unitary Brownian motion $Y =
(Y_t)_{t \geq 0}$ in a $W^{\star}$-non commutative probability space such that
$Y$ and $P$ are $\star$-free in Voiculescu's sense, the main result of this
paper states that $P$ and $Y_tPY_t^{\star}$ are in general position at any time
$t$. To this end, we study the dynamics of the unitary operator
$SY_tSY_t^{\star}$ where $S = 2P-1$. More precisely, we derive a partial
differential equation for the Herglotz transform of its spectral distribution,
say $\mu_t$. Then, we provide a flow on the interval $[-1,1]$ in such a way
that the Herglotz transform of $\mu_t$ composed with this flow is governed by
both the Herglotz transforms of the initial ($t=0$) and the stationary ($t =
\infty)$ distributions. This fact allows to compute the weight that $\mu_t$
assigns to $z=1$ leading to the main result. As a by-product, the weight that
the spectral distribution of the free Jacobi process assigns to $x=1$ follows
after a normalization. In the last part of the paper, we use combinatorics of
non crossing partitions in order to analyze the term corresponding to the
exponential decay $e^{-nt}$ in the expansion of the $n$-th moment of
$SY_tSY_t^{\star}$.
