Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 12 - Review - Concept Check - Page 841: 12

Answer

Use the normal vectors of the plane. Say planes $p_1$ and $p_2$ have normal vectors $n_1$ and $n_2$ respectively. Now find the norm of the normal vectors and find their dot product and use the formula$$ |n_1n_2|=|n_1||n_2|cos\theta$$$$\implies cos\theta=\frac{|n_1n_2|}{|n_1||n_2|}$$ and solve the angle. If the angle $\theta$ is acute or right, then it is the angle between the two planes. If the angle $\theta$ is obtuse, then subtract it from $180^\circ$; then it will be the angle between the two planes.

Work Step by Step

Use the normal vectors of the plane. Say planes $p_1$ and $p_2$ have normal vectors $n_1$ and $n_2$ respectively. Now find the norm of the normal vectors and find their dot product and use the formula$$ |n_1n_2|=|n_1||n_2|cos\theta$$$$\implies cos\theta=\frac{|n_1n_2|}{|n_1||n_2|}$$ and solve the angle. If the angle $\theta$ is acute or right, then it is the angle between the two planes. If the angle $\theta$ is obtuse, then subtract it from $180^\circ$; then it will be the angle between the two planes.
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