Functions Modeling Change: A Preparation for Calculus, 5th Edition

Published by Wiley
ISBN 10: 1118583191
ISBN 13: 978-1-11858-319-7

Chapter 12 - Vectors and Matrices - 12.2 The Components of a Vector - Exercises and Problems for Section 12.2 - Exercises and Problems - Page 509: 16

Answer

$71.57^{\circ}$

Work Step by Step

We see that the initial point is: $(1,5)$. The ending point is: $(2,4)$. So, $\vec{w} =(2-1)\vec{i}+(4-5) \vec{j}=\vec{i}-\vec{j}$ and $\vec{v}=(5-3)\vec{i}+(5-4) \vec{j}=2\vec{i}+\vec{j}$ Now, the angle between two vectors can be computed as: $\cos \theta =\dfrac{\vec{v} \cdot \vec{w}}{|v||w|}=\dfrac{(\vec{i}-\vec{j}) (2\vec{i}+\vec{j})}{(\sqrt {1+1})(\sqrt {4+1})}\\=\dfrac{1}{\sqrt {10}}$ Therefore, $\theta =\cos^{-1}(\dfrac{1}{\sqrt 2})=71.57^{\circ}$
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