Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 13 - Vector Geometry - 13.3 Dot Product and the Angle Between Two Vectors - Exercises - Page 666: 36


$$(v+w)\cdot (v+w)-2v\cdot w = \|v\|^2+\|w\|^2.$$

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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