Question

The non-zero vectors $\overrightarrow{a},\overrightarrow{b}\mathrm{and}\overrightarrow{c}\mathrm{are}\mathrm{related}\mathrm{by}\overrightarrow{a}=8\overrightarrow{b}\mathrm{and}\overrightarrow{c}=-7\overrightarrow{b}$, then the angle between $\overrightarrow{a}\mathrm{and}\overrightarrow{c}$ is ____________.

Answer 1

Given:
$\overrightarrow{a}=8\overrightarrow{b}\mathrm{and}\overrightarrow{c}=-7\overrightarrow{b}$


$$\begin{gathered} \mathrm{Let}\theta \mathrm{be}\mathrm{the}\mathrm{angle}\mathrm{between}\overrightarrow{a}\mathrm{and}\overrightarrow{c}. \\ \overrightarrow{a}·\overrightarrow{c}=\left(8\overrightarrow{b}\right)·\left(-7\overrightarrow{b}\right) \\ \Rightarrow \left|\overrightarrow{a}\right|\left|\overrightarrow{c}\right|\cos \theta =8\times \left(-7\right)\times \overrightarrow{b}·\overrightarrow{b} \\ \Rightarrow \left|8\overrightarrow{b}\right|\left|-7\overrightarrow{b}\right|\cos \theta =-56\times {\left|\overrightarrow{b}\right|}^2 \\ \Rightarrow 56{\left|\overrightarrow{b}\right|}^2\cos \theta =-56\times {\left|\overrightarrow{b}\right|}^2 \\ \Rightarrow \cos \theta =-1 \\ \Rightarrow \theta =\mathrm{\pi } \end{gathered}$$

Hence, the angle between $\overrightarrow{a}\mathrm{and}\overrightarrow{c}$ is $\underline{\mathrm{\pi }}$.