Relation between a set and its orthogonal complement

Question

Here is one fact that i don't seem to understand :

Suppose $V$ is an inner product space and $S$ is a subset of $V$ then if $x \in S$, why is it, that $x$ is orthogonal to every element of $S ^\bot$ ?

I know the definition of $S ^\bot$ i.e. set of all vectors in $V$ that are orthogonal to every vector of a non empty subset $S$ of $V$, but i can't really relate the two things.

Any help will be appreciated.

Answer 1

$S^\perp$ contains exactly those vectors $v$ which are orthogonal to all vectors of $S$.

In particular, for $x\in S$, any $v\in S^\perp$ must satisfy $x\perp v$.

Answer 2

if $x$ is in $S$ then $x$ will be in $S'$ (S perp) only if $x=0$(zero element in $V$).