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 $|\bold{y}|$ are the magnitudes of vector $\bold{x}$ and $\bold{y}$, respectively. This definition is useful to calculate the angle between two vectors.

Magnitude of a vector

The magnitude (also known as length) of a vector, $\bold{v} \in \R^n$, is defined as:

$$|\bold{v}| = \sqrt{\bold{v} \cdot \bold{v}}=\sqrt{\sum_i v_i^2}$$

Example

Consider the vector $\bold{x} =\begin{bmatrix}5 \\ 2\end{bmatrix} \in \R^2$. The magnitude of $\bold{x}$ is

$$|\bold{x}| = \sqrt{5^2+2^2} = \sqrt{25+4}= \sqrt{29}$$

Angle between two vectors

The geometric definition of a dot product can be used to find the angle, $\theta$, between two vectors, $\bold{x}, \bold{y} \in \R^2$, as follows:

$$\theta = \arccos{\frac{\bold{x}\cdot \bold{y}}{|\bold{x}||\bold{y}|}}$$

Example

Consider two vectors, $\bold{x}, \bold{y} \in \R^2$ are $\bold{x}=\begin{bmatrix}5 \\ 2\end{bmatrix}$ and $\bold{y}=\begin{bmatrix}3 \\ -1\end{bmatrix}$, which have the magnitudes, $|\bold{x}|=\sqrt{29}$, $|\bold{y}|=\sqrt{10}$. The angle between them is

$$\theta = \arccos{\frac{13}{\sqrt{290}}}= 40.236$$