Relationship between angle of vectors and orthogonalization

Let $U$ be the subspace of $\mathbb{R}^n$ spanned by $u_1,...,u_d$. Both $a$ and $b$ can be uniquely decomposed into their projections $u_a$, $u_b$ on $U$ and a vector $v_a$, $v_b$ orthogonal to $U$. As $a$ and $b$ are similar (the angle between them is small) their projections are similar.

$v_a$, $v_b$ in their turn are similar.

As the starting points ($u_a$ resp. $u_b)$ and endpoints ($a$ resp. $b$) are close to one another the angle between $v_a$ and $v_b$ is therefore also small.

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