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