Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 12: Vectors and the Geometry of Space - Section 12.3 - The Dot Product - Exercises 12.3 - Page 712: 17


Sum of two vectors having same length will always be orthogonal to their difference.

Work Step by Step

Let us consider two vectors $v_1$ and $v_2$. and $(v_1+v_2) \cdot (v_1-v_2)=v_1 \cdot v_1+v_2 \cdot v_1-v_1 \cdot v_2 -v_2 \cdot v_2$ Therefore, $v_1 \cdot v_1+v_2 \cdot v_1-v_1 \cdot v_2 -v_2 \cdot v_2=|v_1|^2-|v_2|^2$ To get the answer equal to zero we will have to consider $|v_1|^2 =|v_2|^2$. This can be only possible when both have the same length. This implies that $|v_1|^2-|v_2|^2=|v_1|^2-|v_1|^2=0$ we conclude that the sum of two vectors having same length is always orthogonal to their difference.
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