I'm vetting a Linear Algebra book for possible use in a class, and there's an exercise that asks an interesting question. The first task is to show that, for any two vectors $\bf{x}, \bf{y}\in\Bbb{R}^n$, we have:
$$\|\bf{x} + \bf{y}\|^2 + \|\bf{x} - \bf{y}\|^2 = 2\left(\|\bf{x}\|^2 + \|\bf{y}\|^2\right)$$
This is utterly straightforward, and I'm not asking how to show it; you just do the calculation with dot products and that's it. The interesting question is this: The exercise goes on to request a geometric interpretation of this result.
Now, I can come up with geometric interpretations, but I struggle to find anything really satisfying. One approach is to note that $\bf{x}+\bf{y}$ and $\bf{x} -\bf{y}$ are the two diagonals of the parallelogram formed by $\bf{x}$ and $\bf{y}$, and we can then talk about how, if you take the two diagonals, and make them legs of a right triangle, then the square on the hypotenuse will have twice the area it would have had, if we had used the edges of the original parallelogram as legs instead. That seems... rather contrived?
Does anyone know of, or can anyone see, a pleasing, intuitive geometric interpretation of this fairly trivial, quotidian fact about magnitudes of vectors?
The most common interpretation is that the sum of the squared lengths of the diagonals of a parallelogram is equal to the sum of the squared lengths of the sides. Hence the name parallelogram law for this identity.