• :00Days
• :00Hours
• :00Mins
• 00Seconds
A new era for learning is coming soon

Suggested languages for you:

Americas

Europe

Q5E

Expert-verified
Found in: Page 215

### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

# Find the angle ${\mathbit{\theta }}$ between each of the pairs of vectors $\stackrel{\mathbf{\to }}{\mathbf{u}}$ and in exercises 4 through 6.5. $\stackrel{\mathbf{\to }}{\mathbf{u}}{\mathbf{=}}\left[\begin{array}{c}1\\ 2\\ 3\end{array}|\stackrel{\to }{V}=\left[\begin{array}{c}2\\ 3\\ 4\end{array}\right\right]$ .

The angle $\theta$ between $\stackrel{\to }{u}$ and $\stackrel{\to }{v}$ is about 0.1 2 1 9 radians.

See the step by step solution

## Step 1: Correlation coefficient

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

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

${\mathbit{r}}{\mathbf{=}}{\mathbit{c}}{\mathbit{o}}{\mathbit{s}}\left(\theta \right){\mathbf{=}}\frac{\stackrel{\mathbf{\to }}{\mathbf{x}}\mathbf{.}\stackrel{\mathbf{\to }}{\mathbf{y}}}{\mathbf{|}\mathbf{|}\stackrel{\mathbf{\to }}{\mathbf{x}}\mathbf{|}\mathbf{|}\mathbf{.}\mathbf{|}\mathbf{|}\stackrel{\mathbf{\to }}{\mathbf{y}}\mathbf{|}\mathbf{|}}$

## Step 2: Substitute the values into the angle formula

The given vectors are $\stackrel{\to }{u}=\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]$ and $\stackrel{\to }{v}=\left[\begin{array}{c}2\\ 3\\ 4\end{array}\right]$.

The angle formula role="math" localid="1659426804326" $\theta =co{s}^{-1}=\frac{\stackrel{\to }{\mathrm{x}}.\stackrel{\to }{\mathrm{y}}}{||\stackrel{\to }{\mathrm{u}}||.||\stackrel{\to }{\mathrm{v}}||}$.

Find $\stackrel{\mathit{\to }}{\mathit{u}}.\stackrel{\mathit{\to }}{\mathit{v}}$ and $||\stackrel{\mathit{\to }}{\mathit{u}}||.||\stackrel{\mathit{\to }}{\mathit{v}}||$

$\stackrel{\mathit{\to }}{\mathit{u}}\mathit{.}\stackrel{\mathit{\to }}{\mathit{v}}=\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right].\left[\begin{array}{c}2\\ 3\\ 4\end{array}\right]\phantom{\rule{0ex}{0ex}}=1×2+3+3×4\phantom{\rule{0ex}{0ex}}=20$

$||\stackrel{\mathit{\to }}{u}||=\sqrt{{1}^{2}+{2}^{2}+{3}^{2}}⇒\sqrt{14}\phantom{\rule{0ex}{0ex}}||\stackrel{\mathit{\to }}{v}||=\sqrt{{2}^{2}+{3}^{2}+{4}^{2}}⇒\sqrt{29}$

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

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

Hence, the value of $\theta$ is about $0.1219$ radians.