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: 21

Answer

$QPR \approx 48^\circ$, $PQR \approx 75^\circ$, $PRQ \approx 58^\circ$

Work Step by Step

$PQ= \lt 0-2, 3-0 \gt= \lt -2,3 \gt$ $PR= \lt 3-2, 4-0 \gt= \lt 1,4 \gt$ $|PQ|=\sqrt {13}$ and $|PR|=\sqrt {17}$ $PQ \cdot PR= \lt -2,3 \gt \cdot \lt 1,4 \gt=10$ $ \theta = cos^{-1}\dfrac{PQ \cdot PR}{|PQ||PR|}=cos^{-1}\dfrac{10}{\sqrt {221}}\approx 47.726 ^ \circ$ $QP= \lt 2-0, 0-3 \gt= \lt 2,-3 \gt$ $QR= \lt 3-0, 4-3 \gt= \lt 3,1 \gt$ $|QP|=\sqrt {13}$ and $|QR|=\sqrt {10}$ $QP \cdot QR= \lt 2,-3 \gt \cdot \lt 3,1 \gt=3$ $ \theta = cos^{-1}\dfrac{QP \cdot QR}{|QP||QR|}=cos^{-1}\dfrac{3}{\sqrt {130}}\approx 74.745 ^ \circ$ $RP= \lt 2-3, 0-4 \gt= \lt -1,-4 \gt$ $RQ= \lt 0-3, 3-4 \gt= \lt -3,-1 \gt$ $|RP|=\sqrt {17}$ and $|RQ|=\sqrt {10}$ $RQ \cdot RP= 7$ $ \theta = cos^{-1}\dfrac{RQ \cdot RP}{|RP||RQ|}=cos^{-1}\dfrac{7}{\sqrt {170}}\approx 57.529 ^ \circ$ Hence, $QPR \approx 48^\circ$, $PQR \approx 75^\circ$, $PRQ \approx 58^\circ$
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