You should revise your question.
——
You have provided the geometric definition for the dot product (aka inner product) of two vectors. The definition can be accepted straight away. Are you asking to prove the algebraic definition of the dot product is equal to the geometric definition of the dot product?
Algebraic:
+ … +
Geometric:
where = angle between the two vectors
As I thought about it, I figured that must be what you want. Here is the proof:
I will use capital letters for vectors, lower case for scalers.
Also |A| means the magnitude of A, found using ordinary root-of-squares method.
Let be a set of unit vectors in the n-dimensional space.
So
if
if
(1) and
(2) <—<<< Remember and are scalers
and the algebraic dot product is defined as:
(*)
—
Now, use the geometric dot product definition to find :
<—<<< Same as in (1)
(and similarly for )
and can be thought of as the magnitude of the component of A, B that project in the direction, respectively.
and substituting for B:
Finally, using the distributive property of dot products :
<—<<< Same as (*)