How is the following equation proved/rationalized?
A·B = |A| |B| cos(θ)
this comes from the definition of the inner product (dot product)
It 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(q)
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)