Question

For each pair of vectors $\overrightarrow{\mathbf{u}}$and $\overrightarrow{\mathbf{v}}$listed in Exercises 7 through 9, determine whether the angle $\mathit{\theta }$between $\overrightarrow{\mathbf{u}}$and $\overrightarrow{\mathbf{v}}$is acute, obtuse, or right.

8.$\overrightarrow{\mathbf{u}}\mathbf{=}\left[\begin{array}{c}2\\ 3\\ 4\end{array}\right]\mathbf{,}\overrightarrow{\mathbf{v}}\mathbf{=}\left[\begin{array}{c}2\\ -8\\ 5\end{array}\right]$

Short answer

The angle$\theta$between$\overrightarrow{u}$and$\overrightarrow{\mathrm{v}}$is right.

Step-by-step solution

Angle between two vectors

Consider two nonzero vectors $\overrightarrow{\mathbf{x}}$and$\overrightarrow{\mathbf{y}}$ in ${\mathit{R}}^{\mathbf{n}}$.The angle between these vectors is defined as:

role="math" localid="1659428713557" $\mathit{\theta }\mathbf{=}\mathit{a}\mathit{r}\mathit{c}\mathit{c}\mathit{o}\mathit{s}\mathbf{=}\frac{\overrightarrow{\mathbf{x}}\mathbf{.}\overrightarrow{\mathbf{y}}}{\mathbf{\left|}\mathbf{\right|}\overrightarrow{\mathbf{x}}\mathbf{\left|}\mathbf{\right|}\mathbf{.}\mathbf{\left|}\mathbf{\right|}\overrightarrow{\mathbf{y}}\mathbf{\left|}\mathbf{\right|}}$

Find the angle

Now, find the angle by substituting the required values into the formula

$\theta =arccos=\frac{\overrightarrow{\mathrm{u}}.\overrightarrow{\mathrm{v}}}{\left|\right|\overrightarrow{u}\left|\right|.\left|\right|\overrightarrow{v}\left|\right|}$.

$$\begin{gathered} \theta =arccos=\frac{\overrightarrow{\mathrm{u}}.\overrightarrow{\mathrm{v}}}{\left|\right|\overrightarrow{u}\left|\right|.\left|\right|\overrightarrow{v}\left|\right|} \\ =arccos\frac{\left[\begin{array}{c}2\\ 3\\ 4\end{array}\right].\left[\begin{array}{c}2\\ -5\\ 8\end{array}\right]}{\sqrt{\left(2^2+3^2+4^2\right)\sqrt{\left(2^2+{\left(-8\right)}^2+5^2\right)}}} \\ =arccos\frac{2.2-3.8+4.5}{\left(\sqrt{4+9+16}\right)\left(\sqrt{4+64+25}\right)} \\ =arccos\frac{4-24+20}{\sqrt{29.93}} \\ =arccos\frac{0}{\sqrt{2697}} \\ =arc\cos \left(0\right) \\ =90° \end{gathered}$$

Hence, the value of angle$\theta$ is $90°$, and therefore right.