Dot Product vs. Inner Product

In the previous lessons, we described a (typical) vector as a collection of ordered sequences of numbers. In contrast, a generalized vector can be any object that is an element of a vector space. The dot product (also known as the scalar product) is defined for a pair of typical vectors, whereas the inner product is defined for generalized vectors. The result of both operations is a scalar.

Dot product

A dot product, $\bold{x} \cdot \bold{y}$ of two vectors, $\bold{x}, \bold{y} \in \R^n$ may be defined algebraically and geometrically.

Algebraic definition

Algebraically, a dot product is defined as:

$$\bold{x} \cdot \bold{y} = \sum_{j=0}^{n} x_i\times y_i$$

Example

Consider two vectors $\bold{x}, \bold{y} \in \R^2$ are $\bold{x}=\begin{bmatrix}5 \\ 2\end{bmatrix}$ and $\bold{y}=\begin{bmatrix}3 \\ -1\end{bmatrix}$. Their dot product is

$$5\times3+2\times -1=13$$

Geometric definition

Geometrically, a dot product is defined as

$$\bold{x}\cdot \bold{y} = |\bold{x}||\bold{y}|\cos \theta$$

where, $|\bold{x}|$ and $|\bold{y}|$ ...