The dot product can be derived using the law of cosines-
let a, b, c be vectors 

define vector c = a - b 

this forms a triangle, let θ be the angle between a and b, i.e. opposite side c 


from the law of cosines 

c² = a² + b² - 2abcos(θ) 

from another property of the dot product, x·x = x² 

we replace c², a², and b² to get 

c·c = a·a + b·b - 2abcos(θ) 

since, c = a - b 
c·c = (a -b)·(a - b) 
c·c = (a·a - 2a·b + b·b) 

plug this in 

(a·a - 2a·b + b·b) = a·a + b·b - 2abcos(θ) 

clean it up by canceling a·a, and b·b to get: 

-2a·b = -2abcos(θ) 

divide by -2 to get the result 

a·b = abcos(θ)