[JadeNB]

> Then we jump to using something called the Hadamard component-wise product which is not defined.

Aside from the 'Hadamard', which is just the name of a mathematician who used it, it's pretty self documenting, no? Multiply two vectors of the same size by multiplying corresponding components. (This can be viewed as an intermediate, coordinate-dependent step in the computation of the coordinate-independent dot product—although that doesn't seem to be how it's used here.)