Question

Find the angle$\mathit{\theta }$ between each of the pairs of vectors$\overrightarrow{\mathbf{u}}$ and in exercises 4 through 6.

5.$\overrightarrow{\mathbf{u}}\mathbf{=}\left[\left.\begin{array}{c}1\\ 2\\ 3\end{array}\right|\overrightarrow{V}=\left[\begin{array}{c}2\\ 3\\ 4\end{array}\right.\right]$ .

Short answer

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

Step-by-step solution

Correlation coefficient

The correlation coefficientrbetween two characteristics of a population is the

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

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

 Step 2: Substitute the values into the angle formula

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

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

Find $\overrightarrow{\mathit{u}}.\overrightarrow{\mathit{v}}$and $\left|\right|\overrightarrow{\mathit{u}}\left|\right|.\left|\right|\overrightarrow{\mathit{v}}\left|\right|$

$$\begin{gathered} \overrightarrow{\mathit{u}}\mathit{.}\overrightarrow{\mathit{v}}=\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right].\left[\begin{array}{c}2\\ 3\\ 4\end{array}\right] \\ =1\times 2+3+3\times 4 \\ =20 \end{gathered}$$

$$\begin{gathered} \left|\right|\overrightarrow{u}\left|\right|=\sqrt{1^2+2^2+3^2}\Rightarrow \sqrt{14} \\ \left|\right|\overrightarrow{v}\left|\right|=\sqrt{2^2+3^2+4^2}\Rightarrow \sqrt{29} \end{gathered}$$

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

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

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