Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 12 - Vectors and the Geometry of Space - 12.3 The Dot Product - 12.3 Exercises - Page 853: 25


The triangle is a right-angled.

Work Step by Step

Suppose $a=PQ= (2,0,-4)-(1,-3,-2)= \lt 1,3,-2 \gt$ $b=QR=(6,-2,-5)-(2,0,-4)= \lt 4,-2,-1 \gt$ $c=RP=(1,-3,-2)-(6,-2,-5)= \lt -5,-1,3 \gt$ Now we will take dot product of $a$ and $b$. $a \cdot b= \lt 1,3,-2 \gt \cdot \lt 4,-2,-1 \gt=4-6+2=0$ Since the dot product of $a$ and $b$ is $0$ , this implies that $a \perp b$ or $ PQ \perp QR$, so these two side form a right angle. Hence, the triangle is a right triangle.
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