Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 12 - Vectors and the Geometry of Space - 12.3 The Dot Product - 12.3 Exercises - Page 854: 61


$ |a \cdot b|\leq |a| |b|$

Work Step by Step

Theorem $3$ states that: $a \cdot b=|a||b| cos \theta$ .... (1) As $|a| \gt 0$ and $ |b| \gt 0$ , then $|a \cdot b|=||a||b| cos \theta|=|a||b| |cos \theta|$ .... (2) (i) If $ 0 \leq \theta \leq \pi/2$, then If $ 0 \leq |a||b| |cos \theta| \leq |a||b| |$ (ii) If $ \pi/2 \leq \theta \leq \pi$, then If $ |a||b| \geq |a \cdot b| | \gt 0$ From the above two conditions, we have found that $ |a \cdot b|\leq |a| |b|$
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