# What is the difference between the dot product and the element-by-element multiplication?

What is the different between the dot product "$$\cdot$$" and the element-by-element multiplication notation $$\odot$$ in Statistics? I referred to Hamilton's Time-Series Analysis, and these seem to have the same definition. For instance, for two $$n\times 1$$ vectors $$\vec{a}$$ and $$\vec{b}$$, is $$\begin{equation} \vec{a}\odot \vec{b}\overset{?}{\equiv} \vec{a} \cdot \vec{b}=\sum\limits_{i=1}^n a_ib_i \end{equation}$$

$$c = \vec{x}^T \cdot \vec{y} = \begin{bmatrix} x_{1} \\ x_{2} \\ \vdots \\ x_{n} \end{bmatrix}^T \cdot \begin{bmatrix} y_{1} \\ y_{2} \\ \vdots \\ y_{n} \end{bmatrix} = [x_1, x_2, \cdots, x_n] \cdot \begin{bmatrix} y_{1} \\ y_{2} \\ \vdots \\ y_{n} \end{bmatrix} = \sum_{i=1}^n x_iy_i \in \mathbb{R}$$ $$\vec{z} = \vec{x} \odot \vec{y} = \begin{bmatrix} x_{1} \\ x_{2} \\ \vdots \\ x_{n} \end{bmatrix} \odot \begin{bmatrix} y_{1} \\ y_{2} \\ \vdots \\ y_{n} \end{bmatrix} = \begin{bmatrix} x_1y_{1} \\ x_2y_{2} \\ \vdots \\ x_ny_{n} \end{bmatrix} \in \mathbb{R}^n$$
Note that $$\vec{x}, \vec{y}$$ must have the same dimension. The Hadamard product is often defined in terms of matrices in many sources, but for general tensors it suffices that they have the same shape. For two vectors, this is tantamount to having the same dimension.
So in general $$\vec{x} \cdot \vec{y} \neq \vec{x} \odot \vec{y}$$.
Furthermore, $$\|\vec{x} \odot \vec{y}\| \leq \|\vec{x}\|\|\vec{y}\|$$ and $$\|\vec{x} \cdot \vec{y}\| \leq \|\vec{x}\| \|\vec{y}\|$$. But since $$Pr\left( \|\vec{x} \cdot \vec{y}\| \leq \|\vec{x} \odot \vec{y}\| \right) \approx .67$$ for some choice of probability space (see link on $$\|\vec{x} \odot \vec{y}\| \leq \|\vec{x}\|\|\vec{y}\|$$), there isn't a direct inequality between their norms.