## Proving Σ(u_i v_i) for i = 1,2,... & ||u|| ||v|| cos γ equivalence (using rota. mat.)

Here is the question to which I am referring, along with its solution.:
Question #3:
https://www.docdroid.net/CtqTsdX/e2.pdf

Question #3's Solution:
https://www.docdroid.net/CtqTsdX/e2.pdf#page=3

Is the only important (generally-applicable) part of the proof the "u ⋅ v = (Ru) ⋅ (Rv) = u^T R^T Rv = ||u|| ||v|| cos γ" part? That is, is the "Consider a rotation matrix R that rotates u to (||u||, 0, 0) and that rotates v to ||v||(cos γ, sin γ, 0)." a specific example (that is not a general statement) (used for attempting to help the reader better understand the presented situation)? If I'm wrong, could someone please elaborate? Also, how does that prove the equivalence between the sum of u_i v_i for i = 1,2,3, ... and ||u|| ||v|| cos γ?

Any input would be GREATLY appreciated!

P.S.
Slides of "lecture 2" (should they be deemed necessary or at least useful):
https://www.docdroid.net/LPEdvTJ/2-s...9pp.pdf#page=2