[Math] Basic confusion about direct sums and orthogonality

Question

Say I have a vector space has a not necessarily orthogonal direct sum
decomposition $V = A \oplus B \oplus C$, which means that
every $v \in V$ has a unique decomposition $v = a + b + c$, where
$a \in A$, $b \in B$ and $c \in C$. Suppose $A = {\rm span}\left\{a\right\}$. Now, let $v^{\perp_A}$ represent the
component of $v$ perpendicular to $A$, that is,
$$ v^{\perp_A} = v – \langle a, v\rangle a/||a||^2.$$
Therefore, $v^{\perp_A}$ necessarily $\in B\oplus C$ since $v^{\perp_A} = 0 + v^{\perp_A}$, where $0 \in A$. Since $v = v^{\perp_A} + (v – v^{\perp_A})$ is valid decomposition for any $v$ and unique, $B\oplus C$ and $A$
are perpendicular subspaces. In other words, if a subspace in a direct
sum is spanned by only 1 vector, it must be perpendicular to all other
subspaces in the direct sum. Please let me know if that is wrong!

Answer 1

Hint: $$v^{\bot_A} = v - \frac {\langle a, v \rangle}{\lVert a \rVert^2} a = (a + b + c) - \frac {\langle a, v \rangle}{\lVert a \rVert^2} a = \left (1 - \frac {\langle a, v \rangle}{\lVert a \rVert^2} \right ) a + b + c.$$