There are many possible inner products, but the most important, for vectors with entries in \(\mathbb{C}\) is the usual dot product:
\begin{equation*} (u,v) := u^\star v = \sum_{i=0}^n \bar{u}_i v_i. \end{equation*}For real-valued vectors it can be understood as the “angle” between the vectors in \(\mathbb{R}\text{.}\) That is, if \(\alpha\) is the angle between two vectors, \(u\) and \(v\text{,}\) then,
\begin{equation*} \cos(\alpha) = \frac{(u,v)}{\sqrt{(u,u)} \sqrt{(v,v)}}. \end{equation*}The most important/interesting situation when \((u,v)=0\text{,}\) in which case we say that \(u\) and \(v\) are orthogonal.
in-context