Linear Algebra: A Modern Introduction

Published by Cengage Learning
ISBN 10: 1285463242
ISBN 13: 978-1-28546-324-7

Chapter 1 - Vectors - 1.2 Length and Angle: The Dot Product - Exercises 1.2 - Page 290: 30


The proof is below.

Work Step by Step

We need to prove $\triangle ABC$ is a right triangle. In order to this, we will have to find a right angle.When the dot product for two vectors is zero, they form a right angle. Let us take three vectors $\overrightarrow{AB},\overrightarrow{AC},\overrightarrow{BC}$ and find their dot product. Now, $\overrightarrow{AB} \cdot \overrightarrow{AC}=4(7)+(-2)(4)=20$ and $\overrightarrow{AB} \cdot \overrightarrow{BC}=(4)(3)+(-2)(6)=0$ We can see that that the vectors $\overrightarrow{AB}$ and $ \overrightarrow{BC}$ forms a right angle. Thus, $\angle ABC=90^{\circ}$
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