How to prove invariance of dot-product to rotation of coordinate system Using the definition of a dot-product as the sum of the products of the various components, how do you prove that the dot product will remain the same when the coordinate system rotates? 
Preferably an intuitive proof please, explainable to a high-school student. Thanks in advance.

First you should show that for any two vectors $v$ and $w$ in $\mathbb{R}^n$ (taking $n=3$ if necessary) $v\cdot w = |v||w|\operatorname{cos}\theta $, where $\theta$ is the (smaller) angle between both vectors. 
This is a very geometric fact and you can probably prove it to them if they know the cosines law. First observe that:
$$||v-w||^2 = ||v||^2+||w||^2-2||v||w||\operatorname{cos}\theta.$$
This comes from the the fact that the vectors $v$, $w$ and $(v-w)$ form a triangle (draw it). On the other hand,
$$||v-w||^2 = (v-w)\cdot(v-w)=v\cdot v -2v\cdot w + w\cdot w .$$
The result follows immediately. After this you only need to observe that rotations don't affect lenghts or angles, then by the formula above the dot product is invariant under rotations.
