Let $X = \mathbb{R}^{n}$. Known there is an orthogonal basis $\{u, q_1, q_2,...,q_n\}$ for $X$, but only the first basis vector $u$ is given explicitly. Let $v$ be an arbitrary vector in $X$ with respect to above basis.
Now, I want to find the projection of $v$ onto the orthogonal complement of $u$, which should be $u^{⊥}=span\{q_1, q_2,...,q_n\}$, however, I don't want to compute the orthogonal complement of $u$ explicitly. So, am I be able to find the projection of $v$ onto $u^{⊥}$ by doing following
$$ v -\frac{(u^Tv)u}{||u||^2} $$
which is subtracting the projection of $v$ onto $u$ from $v$ ? This is clear in 2D graph, but I am not sure this is still valid for higher dimension.