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Q16E

Expert-verified

Essential Calculus: Early Transcendentals

Found in: Page 556

Book edition 2nd

Author(s) James Stewart

Pages 830 pages

ISBN 9781133112280

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Short Answer

To find the angle between vectors \(a\) and \(b\) vectors.

The exact angle between two vectors a and \({\rm{b}}\) is \(\theta = {\cos ^{ - 1}}\left( {\frac{4}{5}} \right)\).

The approximate angle between two vectors a and \({\rm{b}}\) is \(\theta \approx {37^\circ }\).

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Concept of Dot Product of two vectors

Formula used:

1. The magnitude of a vector \({\bf{a}} = \left\langle {{a_1},{a_2},{a_3}} \right\rangle \) is \(|{\rm{a}}| = \sqrt {a_1^2 + a_2^2 + + a_3^2} \).

2. The dot product of two vectors \({\rm{a}}\) and \({\rm{b}}\) in terms of \(\theta \) is \({\rm{a}} \cdot {\rm{b}} = |{\rm{a}}||{\rm{b}}|\cos \theta \).

Step 2: Calculation of the dot product of two vector \(a \cdot b\)

The given two vectors are \({\rm{a}} = \langle 4,0,2\rangle \) and \({\rm{b}} = \langle 2, - 1,0\rangle \).

Consider a general expression to find the dot product between three-dimensional vectors.

\(\begin{aligned}{l}{\rm{a}} \cdot {\rm{b}} &= \left\langle {{a_1},{a_2},{a_3}} \right\rangle \cdot \left\langle {{b_1},{b_2},{b_3}} \right\rangle \\{\rm{a}} \cdot {\rm{b}} &= {a_1}{b_1} + {a_2}{b_2} + {a_3}{b_3}\end{aligned}\)

Substitute the value of components \({a_1} = 4,{a_2} = 0,{a_3} = 2,{b_1} = 2,{b_2} = - 1\), and \({b_3} = 0\) in the above result.

\(\begin{aligned}{l}{\rm{a}} \cdot {\rm{b}} &= (4)(2) + (0)( - 1) + (2)(0)\\{\rm{a}} \cdot {\rm{b}} &= 8 + 0 + 0\\{\rm{a}} \cdot {\rm{b}} &= 8\end{aligned}\)

Step 3: Calculation of the magnitude of vector \(a\)and \(b\)

Calculate the magnitude of the vector a by Substituting \({a_1} = 4,{a_2} = 0\), and \({a_3} = 2\) in the formula \(\left( 1 \right).\)

\(\begin{aligned}{l}|{\rm{a}}| &= \sqrt {{{(4)}^2} + {0^2} + {2^2}} \\|{\rm{a}}| &= \sqrt {16 + 4} \\|{\rm{a}}| &= \sqrt {20} \\|{\rm{a}}| &= 2\sqrt 5 \end{aligned}\)

Calculate the magnitude of the vector b by Substituting \({b_1} = 2,{b_2} = - 1\), and \({b_3} = 0\) in the formula\(\left( 1 \right).\)

\(\begin{aligned}{l}|{\rm{b}}| &= \sqrt {{2^2} + {{( - 1)}^2} + {0^2}} \\|{\rm{b}}| &= \sqrt {4 + 1} \\|{\rm{b}}| &= \sqrt 5 \end{aligned}\)

Most popular questions for Math Textbooks

Determine the vector equation for the line segment from\((2, - 1,4)\)to\((4,6,1).\)

Find the magnitude of the torque about \(P\) if a \(36 - lb\)force is applied as shown.

To find: The volume of the parallelepiped determined with adjacent edges PQ, PR and PS.

To show

(a) \({\rm{i}} \cdot {\rm{j}} = 0,{\rm{j}} \cdot {\rm{k}} = 0\) and \({\rm{k}} \cdot {\rm{i}} = 0\).

(b) \({\rm{i}} \cdot {\rm{i}} = 1,{\rm{j}} \cdot {\rm{j}} = 1\) and \({\rm{k}} \cdot {\rm{k}} = 1\)

Determine the cross-product between\(a\)and\(b\)and sketch \(a,b\)and\(a \times b\)as vectors starting at the origin.

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