I have been using the definitions of Dot and Cross products to understand several natural phenomena. What I have not yet understood is that how are these products defined the way they are. Having no idea about how they came about only makes me think that these calculations are miracles.
"They surprisingly work!"
I would like to drift away from my surprise because I am bothered by simple acceptance of the product rules. So, how are these products defined?
The usage of the dot and cross products in physics arises from the need to formalize two geometric concepts: projecting vectors onto a line, and producing vectors normal to a surface.
Dot product
In a vector space, one vector can be mapped onto a line parallel to another vector, giving a number that is the length of the "shadow" of the first vector along the second vector, multiplied by the length of the second vector. The dot product defines the process of projection in Euclidean space. For example, in the formula for work
$$W = \int \vec{F}\cdot d\vec{s}$$
we use the dot product to add up only those components of the force $\vec{F}$ that point along our path $d\vec{s}$.
Cross product
The cross product between two vectors produces a vector that is normal to a surface. There are many ways in which this can be done (for another example, cf. Gram-Schmidt orthogonalization: https://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process). This operation has two advantages over other methods: firstly, it is often easier, being a one-step process rather than requiring multiple calculations. Secondly, the magnitude of the cross product is the area of the parallelogram formed by the two operand vectors. This means that the magnitude of the cross product is also a kind of projection, in that it projects one vector onto a line coplanar with both vectors and orthogonal to the other vector. For example, revolving objects have a linear velocity that is normal to the plane defined by the axis of rotation and the radius vector, and therefore, should be able to be defined as proportional to a cross product of the two. In physics we set this proportionality constant to 1, so that
$$\vec{v} = \vec{r}\times\vec{\omega}$$
Side note on the cross product
The dot product can be defined in any-dimensional Euclidean space, but the cross product is peculiar. It can only be consistently defined in three dimensions*, mainly because of the existence of a field called the quaternions in four-dimensional space. Quaternions are a four-dimensional non-commutative extension of the complex numbers that have a structure (see https://en.wikipedia.org/wiki/Quaternion) that permits quaternions to be directly multiplied, much like ordinary complex numbers. Due to the particular structure of the quaternions, the following identity holds, for three-dimensional vectors $\vec{u}$ and $\vec{v}$:
$$(0,\vec{u})(0,\vec{v})=(-\vec{u}\cdot\vec{v},\vec{u}\times\vec{v})$$
So quaternions combine both the dot and the cross product into one operation. The structure that makes this possible is only logically consistent in four dimensions*.
*There is an eight-dimensional extension of the complex numbers called the octonions, meaning that there is a seven-dimensional analogue of the cross product, but most of the useful properties of the cross product are lost because the structure of the octonions differs somewhat from that of the quaternions under multiplication.
Dot products and cross products give the result they do because they were derived in that way and we thus get the resultant product produced as it comes from the mathematical derivation.
Check out the video of derivation of the dot product and cross product in relation to angles and each other and thus underlying principle will be a lot clearer then:
So, if you want to know what these two products are you can just find them in any Calculus book.
But to make a short answer, the way you can imagine the two products is like this: Imagine you have two vectors $\textbf{A}$ and $\textbf{B}$ which are in a $2D$ space (straight arrows which have an angle $\theta$ between them). The dot product between A and B give you the projection of A on B.
The cross product of A and B on the other hand, gives you a vector C that is perpendicular to both A and B, with a direction given by the $\textbf{right-hand rule}$ and a magnitude equal to the area of the parallelogram that the vectors span.
Of course, there is also an algebraic definition of the dot and cross products. You can look this up even on Wikipedia.
Imagine inventing thousands of arbitrary mathematical methods. Two of them turn out to be tremendously useful. We call them the dot product and the cross product.
Other methods also turn out to be useful. Differentiation, integration, matrixes, linear algebra, tensors, differential equations, eigenvector math, vectors of course, vector fields, ... And so on. Many have names and/or wide ranges of applications as well.
You can pick'n'choose from the methods and tools invented by math. They are not the only ones out there, the dot and cross products just happen to work well in the cases you are dealing with.
See also this other answer of mine where a similar question regarding the cross-product alone is explained.
