Dot Products via Projections

Since we are familiar with the ideas of adding vector sand scaling up vectors by any constant numbers, let us take a look at how we multiply vectors. When we do this, we use what is called the "dot product." Let us define a vector for, if we may say, $n$ number of "dimensions." We write the vector as a set of its components in the various dimensions:

$$\vec{v}_a=[a_1, a_2, a_3,...,a_n]$$

Let us also define another vector, $\vec{v}_b$, with which we may take the dot product of $\vec{v}_a$. So, now we have:

$$\vec{v}_a=[a_1, a_2, a_3,...,a_n]$$ $$\vec{v}_b=[b_1, b_2, b_3,...,b_n]$$

When we take the dot product, the notation we use is, not surprisingly, a dot. To take the dot product of the two vectors we have, we would write:

$${\vec{v}_a}\cdot{\vec{v}_b}$$

The result of this particular dot product would end up being:

$${\vec{v}_a}\cdot{\vec{v}_b}= a_1b_1+a_2b_2+a_3b_3+...+a_nb_n$$

If we take a closer look at the result, we see that the dot product we now have no longer has any direction. It has become a scalar quantity. Is there another property of vectors that has a scalar quantity? Indeed there is. The length of a vector is a scalar as well, and there is a way we can relate it with the dot product. In order to do so, let us take the dot product of a vector with itself. For the sake of simplicity, we will simply write the vector as $A$ instead of ${\vec{v}_a}$.

$$A \cdot A=a_1a_1+a_2a_2+a_3a_3+...+a_na_n=a_1^2+a_2^2+a_3^2+...+a_n^2$$

From the above calculation, we come to the conclusion that the dot product of a vector with itself is simply the sum of its components squared. For vectors, we define length in such a manner, with $\Vert A\Vert$ being the notation for a vector's length:

$$\Vert A\Vert = \sqrt{a_1^2+a_2^2+a_3^2+...+a_n^2}$$

Comparing this to the dot product we had for $a$ with itself earlier, we can relate the two quantities in the following manner:

$$\Vert A\Vert = \sqrt{A \cdot A}$$ or,

$$\Vert A\Vert^2 = {A \cdot A}$$

One of the more useful applications of dot products in Euclidian space is the ability to project a component of one vector along another vector. This is often referred to as scalar projection. Let us consider the example of two vectors, $A$ and $B$, pointing in different directions, separated by an angle of $\theta$. In order to find the projection of $A$ along $B$, we would use the following:

$$A \cdot B = \Vert A\Vert \Vert B\Vert \cos\theta$$

The dot product of a vector with itself is the simplest form of the dot product that will be encountered. Also note that some of the preliminary representations of a dot product are in a very general sense, meaning that there will rarely be more than two or three dimensions that will need to be factored in for a dot product. Dot products have many applications relating to other fields of science, particularly physics. Many formulas, such as the ones for kinetic energy and work, are traditionally defined with dot products, but have been simplified situationally. It is a very important and useful concept to have in mind, as it allows for a bit more intuition for vectors and how they interact with one another.