Why is Euclidean inner product the same as the dot product in orthonormal Cartesian coordinates?

We went through a similar exercise for your question on length and the orthonormal basis. Perhaps we can follow a similar approach to address this question.

Let's suppose we are working with an orthonormal basis spanned by b_1 and b_2. (I'll stick with 2D here, but this can be generalized to n-dimensional vectors.) Then we can represent our vectors as follows: u = u_1 * b_1 + u_2 * b_2 and v = v_1 * b_1 + v_2 * b_2.

The Euclidean inner product is gives <u,v> = u_1*v_1*<b_1,b_1> + u_1*v_2*<b_1,b_2> + u_1*v_1*<b_2,b_1> + u_2*v_2*<b_2,b_2>. Only when <b_1,b_2> = 0 does this expression reduce down to the dot product of two vectors.

Here, I'm using the the elementary properties of the inner product to show that it is equivalent to the dot product when working with an orthonormal basis. This is independent from the equation $\langle u,v \rangle = |u||v| cos(\theta)$, which can be proved separately using the law of cosines.