I might be missing something, but from what I have learned in linear algebra, for a matrix or vector multiplication to be valid you need to have number of columns in pre-multiplying vector to be the same as the number of rows of the multiplies one, i.e. in $\vec{a} \cdot \vec{b}$, $\vec{a}$ should have the number of columns that is equal to the number of rows in $\vec{b}$.
Yet Euclidean norm is defined as:
$$||x|| := \sqrt{x \cdot x}$$
suppose $x$ is a $2$ by $1$ vector. Then you would be multiplying 2 by 1 vector by 2 by 1 vecotr, which is undefined. I must be missing something.
Definition of dot product of $n\times 1$ real matrices:
$$x\cdot y \stackrel{def}= x^Ty$$