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: 32


$$\langle -2,1,0 \rangle.$$

Work Step by Step

Assume that the vector $\langle a, b,c \rangle $ is orthogonal to $\langle 1,2,1 \rangle $, then we have $$\langle a,b,c \rangle \cdot \langle 1,2,1\rangle=0\Longrightarrow a+2b+c=0 .$$ We can pick any vector $\langle a,b,c \rangle$ that satisfies the above equation. Hence we can choose a vector as follows $$\langle -2,1,0 \rangle.$$ One can see that $\langle -2,1,0 \rangle $ is orthogonal to $\langle 1,2,1 \rangle $ and not orthogonal to $\langle 1,0,-1 \rangle $ because their dot product would not be 0.
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