University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 11 - Section 11.3 - The Dot Product - Exercises - Page 616: 19


$ \theta =\dfrac{\pi}{2}$ or $90 ^{\circ}$; $\overrightarrow{CA} \perp\overrightarrow{DB}$

Work Step by Step

We have two diagonals, let us say $\overrightarrow{CA}$ and $\overrightarrow{DB}$, of a rhombus ABCD . The formula to calculate the angle between two diagonals $\overrightarrow{CA}$ and $\overrightarrow{DB}$ of a rhombus is: $ \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}|})$ As ABCD is a rhombus, thus $|\overrightarrow{DA}|=|\overrightarrow{AB}|$ or, $\theta=\cos ^{-1} (0)$ or, $ \theta =\dfrac{\pi}{2}$ or $90 ^{\circ}$ Hence, $\overrightarrow{CA} \perp\overrightarrow{DB}$
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