Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

Find the angle $θ$ between each of the pairs of vectors $\vec{u}$ and in exercises 4 through 6.
5. $\vec{u} = [\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}| \vec{V} = [\begin{matrix} 2 \\ 3 \\ 4 \end{matrix}]$ .

The angle $θ$ between $\vec{u}$ and $\vec{v}$ is about 0.1 2 1 9 radians.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Correlation coefficient

The correlation coefficient r between two characteristics of a population is the

Cosine of the angle between the deviation vectors $\vec{x}$ and $\vec{y}$ for the two characteristics:

$r = c o s (θ) = \frac{\vec{x} . \vec{y}}{| | \vec{x} | | . | | \vec{y} | |}$

Step 2: Substitute the values into the angle formula

The given vectors are $\vec{u} = [\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}]$ and $\vec{v} = [\begin{matrix} 2 \\ 3 \\ 4 \end{matrix}]$ .

The angle formula role="math" localid="1659426804326" $θ = c o s^{- 1} = \frac{\vec{x} . \vec{y}}{| | \vec{u} | | . | | \vec{v} | |}$ .

Find $\vec{u} . \vec{v}$ and $| | \vec{u} | | . | | \vec{v} | |$

$\vec{u} . \vec{v} = [\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}] . [\begin{matrix} 2 \\ 3 \\ 4 \end{matrix}] = 1 \times 2 + 3 + 3 \times 4 = 20$

$| | \vec{u} | | = \sqrt{1^{2} + 2^{2} + 3^{2}} \Rightarrow \sqrt{14} | | \vec{v} | | = \sqrt{2^{2} + 3^{2} + 4^{2}} \Rightarrow \sqrt{29}$

Now, calculate the angle by substituting the required values into the angle formula.

$\begin{array}{l} θ = \cos^{- 1} \frac{20}{\sqrt{14} . \sqrt{29}} \\ \approx 0.1219 \end{array}$

Hence, the value of $θ$ is about $0.1219$ radians.

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Linear Algebra With Applications

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Short Answer

Find the angle $θ$ between each of the pairs of vectors $\vec{u}$ and in exercises 4 through 6.
5. $\vec{u} = [\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}| \vec{V} = [\begin{matrix} 2 \\ 3 \\ 4 \end{matrix}]$ .

Step by Step Solution

Step 1: Correlation coefficient

Step 2: Substitute the values into the angle formula

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Linear Algebra With Applications

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Short Answer

Find the angle θ between each of the pairs of vectors u→ and in exercises 4 through 6.5. u→=[123V→=234] .

Step by Step Solution

Step 1: Correlation coefficient

Step 2: Substitute the values into the angle formula

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Pure Maths

Find the angle $θ$ between each of the pairs of vectors $\vec{u}$ and in exercises 4 through 6.
5. $\vec{u} = [\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}| \vec{V} = [\begin{matrix} 2 \\ 3 \\ 4 \end{matrix}]$ .