The dot product represents the actual “multiplication overlap” between the vectors. On its own it’s just a number, and we often need to compare it to the max possible overlap.

Let’s say we have vector (3,0). It’s length is 3 (technically, sqrt(3^2 + 0^2) = sqrt(9) = 3).

Now we have another vector (4,0). It’s length is 4.

The dot product of (3,0) and (4,0) is 12. This means the vectors have reached their max potential overlap (of 12).

In the case of (2,2) and (1,2) we would say:

(2, 2) has a magnitude of sqrt(2^2 + 2^2) = sqrt(8) = 2.82
(1,2) has a magnitude of sqrt(1^2 + 2^2) = sqrt(5) = 2.23

Max overlap = 2.82 * 2.23 = 6.32
Actual overlap = 6

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Yes, the dot product can represent the cosine if the magnitudes of each of the vectors are 1. From the formula,

a dot b = |a||b| cos(theta) = cos(theta)

So the dot product becomes a simple way to find the cosine of the angle between them using plain arithmetic, no fancy trig functions.

But what number 6 represents? If the dot product only represents how two vectors is pointing the same direction then why we cant just using cosine between angles to represents that. For example 1 means that two vectors points in the same direction, 0 means that vectors are orthogonal and -1 means that vectors points in the oposite direction. Why do we need to multiply cosine by |a| and |b| to represent difference between vectors direction instead of just using cosine between them?
Sorry for my english im not native speaker.