The dot product is a property of real vector spaces which is a simplified version of an inner product; specifically, it obviates the need to complex-conjugate anything because, well, \(\bar{n} = n, n \in \mathbb{R}\). The dot-product also yield a real number.

## constituents

- \(x, y \in \mathbb{R}^{n}\) (NOTE the realness)
- where, \(x = (x_1, \dots, x_{n})\) and \(y = (y_1, …, y_{n})\)

## requirements

As we are familiar with, element-wise product and sum

\begin{equation} x\cdot y = x_1y_1 + \dots + x_{n}y_{n} \end{equation}

## additional information

### properties of the dot product

- For fixed \(y \in \mathbb{R}^{n}\), the dot product map that sends \(x\) to \(x \cdot y\) is linear (inheriting add. and homo. from algebra)
- \(x \cdot x = 0\) IFF \(x =0\) (no negs allowed (above), so every slot has to have a zero to multiply to 0)
- \(x \cdot x > 0\) for all \(x \in \mathbb{R}^{n}\) (neg times neg is pos)
- \(x \cdot y = y \cdot x\) for reals; by inheriting from each element’s field

### orthogonality test

The dot product is an orthogonality test. If the dot product between the two vectors is \(0\), they are definitely orthogonal.

### geometric interpretation of the dot product

Well, we have some shape between two vectors; then, we can first write out the law of cosines. Then, we can see that, for two vectors from the same origin, we can say that the projection of vector \(\vec{A}\) onto \(\vec{B}\) is written as:

\begin{equation} |\vec{A}||\vec{B}|\cos \theta \end{equation}

where, \(\theta\) is the angle between the two vectors.