Dot Product

Vector multiplication is an often-encountered operation that is useful and needs to be performed. There are different kinds of multiplication operations, on this page, we will only cover the dot product. The dot product is a type of inner product, where the inner product is a more general term and the dot product is the term for finite Euclidean spaces.

In a mathematical way, the dot product can be described by the following expression

The further meaning of this expression can be described by the sum of all values of the vectors $\vec{a}$ and $\vec{b}$, starting at the first element of the vector and ending at the last element. In this case, we assume that the vectors are of length n, as can be seen at the top of the summation sign.

Let's work the expression from above out by using a simple example where the two vectors are defined as

By only using the coefficients in front of the unit terms $\vec{e}_x$ and $\vec{e}_y$. The dot product of these two vectors can then be computed and results in the following summation

Geometric visualization

Now the mathematical representation of the dot product is given, it is time to give a clear visualization of what the dot product actually means in a geometrical way. In general, the dot product is encountered in different situations where the interpretation of it can differ. However, in this section, we will show what it looks like geometrically speaking.

Let's consider the vectors $\vec{a}$ and $\vec{b}$ in figure 1. The dot product for geometric purposes is written differently. Now it takes the form of the following expression

In this expression is $\theta$ the angle between the two vectors and $\|a\|$ the norm of the vector a, which equals $\|a\| = \sqrt{a_1 + a_2 + \cdots + a_n}$ and is a scalar value.

Figure 1. Vectors $\vec{a}$ and $\vec{b}$ with an unknown angle $\theta$

By rewriting the expression in equation 4 into a function of only the vectors $\vec{a}$ and $\vec{b}$, the angle $\theta$ can then be computed. This expression is shown below

Example

Let's now compute the angle between the two vectors in figure 1. In this figure the vector $\vec{a}$ and $\vec{b}$ are

substituting this into equation 5 we can compute the angle $\theta$. This results into;

Now by taking the $cos^{-1}$ of the fraction from above, the final angle $\theta$ can be computed in this case, it is computed in degrees.