## Calculus (3rd Edition)

$$(v+w)\cdot (v+w)-2v\cdot w = \|v\|^2+\|w\|^2.$$
Using the properties of the dot product, we have $$(v+w)\cdot (v+w)-2v\cdot w\\ =v\cdot v +v\cdot w+w\cdot v+w\cdot w-2v\cdot w\\ =v\cdot v+w\cdot w= \|v\|^2+\|w\|^2.$$