Why the condition of positive definiteness only depend on one single vector in the inner product?
Positive Definite Matrices
$$K\mapsto(\vec v,\vec w)=\vec v^Tk\vec w\\ \text{Pos.Def.}:(\vec v,\vec v)=\vec v^Tk\vec v=||\vec v||^2$$
Only depends on $v$ here not $w$.
Essentially, definiteness is a property we attribute to bilinear forms, but rather to quadratic forms. As it turns out, a quadratic form $Q(v)$ is derived from a symmetric bilinear form $B(v,w)$ where both arguments are the same vector.