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: 19


$\overrightarrow{CA} \perp\overrightarrow{DB}$

Work Step by Step

Here, ABCD is a rhombus having two diagonals abbreviated as $\overrightarrow{CA}$ and $\overrightarrow{DB}$. The angle between two diagonals of a rhombus ABCD is calculated as: $\overrightarrow{CA}$ and $\overrightarrow{DB}$ $ \theta = \cos ^{-1} (\dfrac{\overrightarrow{CA} \cdot \overrightarrow{DB}}{|\overrightarrow{CA}||\overrightarrow{DB}|})=\cos ^{-1} (\dfrac{(\overrightarrow{DA})^2-(\overrightarrow{AB})^2}{ |\overrightarrow{CA}||\overrightarrow{DB}|})$ Since, ABCD is a rhombus, then $|\overrightarrow{DA}|=|\overrightarrow{AB}|$ or, $\theta=\cos ^{-1} (0)=\dfrac{\pi}{2}$ or $90 ^{\circ}$ This implies that $\overrightarrow{CA} \perp\overrightarrow{DB}$
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