On this answer the hyperplane, presumably in a perceptron classifier, is described as the dot product $\langle \vec{w_{x}},\vec{x} \rangle$, where $\vec{w_x}$ is presumably the vector of weights, and $\vec x$ an example in the training set.
My very tentative understanding is the the hyperplane was the plane defined by the examples in the training set, (or possibly the vector of weights). In other words, a vectorial or geometric (hyperplane) object, rather than a scalar.
Can I get an explanation? Thank you in advance.
The hyperplane is defined by the equation $\langle \vec{w},\vec{x} \rangle = 0$. This hyperplane partitions the training set into two sets, $\{\vec x\mid \langle\vec{w},\vec{x}_i\rangle >= 0\}$, and $\{\vec x\mid \langle\vec{w},\vec{x}_i\rangle <0\}$.