Why does inner product of 2 vectors look like that?

The inner product of two vectors a and b is defined by

a⋅b = a₁b₁ + a₂b₂

where a = (a₁,a₂) and b = (b₁,b₂) are defined on a two-dimensional space.

Why does the inner product of these two vectors look like that? Is there any purpose at all to define the inner product like that?

The answer to this question shows the connection between vectors and geometry. More precisely, the inner product helps us obtain the angle between two vectors.

But there are two angles between these two vectors. The first angle is defined when we stack the two vectors one after the other. The symbols |a| and |b| mean the lengths of vectors a and b.

The second angle is defined when we coincide the starting points of the two vectors.

The two angles, α and γ, are actually connected when we look at the diagram below.

That is, their sum is equal to 180°. From the definition of the inner product, we can conclude right away that a⋅b = 0 if the two vectors are perpendicular to each other.