  We show that if we have an orthogonal base ($u_1,\ldots,u_p$) in a
$p$-dimensional vector space, and select $p+1$ vectors $v_1,\ldots, v_p$ and
$w$ such that the vectors traverse the origin, then the probability of $w$
being to closer to all the vectors in the base than to $v_1,\ldots, v_p$ is at
least 1/2 and converges as $p$ increases to infinity to a normal distribution
on the interval [-1,1]; i.e., $\Phi(1)-\Phi(-1)\approx0.6826$. This result has
relevant consequences for Principal Components Analysis in the context of
regression and other learning settings, if we take the orthogonal base as the
direction of the principal components.
