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 12.5 - Lines and Planes in Space - Exercises 12.5 - Page 727: 49


$1.76 $ rad.

Work Step by Step

The angle between intersecting planes is the angle between their normal vectors. ${\bf n_{1}}=\langle 2,2,2 \rangle, \quad {\bf n_{2}}=\langle 2,-2,-1 \rangle$ $\displaystyle \theta=\cos^{-1}\left( \frac{{\bf n_{1}}\cdot{\bf n_{2}}}{|{\bf n_{1}}|\cdot|{\bf n_{2}}|} \right)=\cos^{-1}(\frac{4-4-2}{\sqrt{4+4+4}\cdot\sqrt{4+4+1}})$ $=\displaystyle \cos^{-1}(\frac{-2}{2\sqrt{3}\cdot 3})$ $=\displaystyle \cos^{-1}(\frac{-\sqrt{3}}{3\cdot 3})$ $\approx 1.76$
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