Question

Find the magnitude of each of the two vectors $\overrightarrow{a}\mathrm{and}\overrightarrow{b}$, having the same magnitude such that the angle between them is 60° and their scalar product is $\frac{9}{2}.$

Answer 1

Let the two vectors be $\overrightarrow{a}\mathrm{and}\overrightarrow{b}$.
Since the vectors have same magnitude so,
$\left|\overrightarrow{a}\right|=\left|\overrightarrow{b}\right|$
Scalar product of two vectors = $\overrightarrow{a}·\overrightarrow{b}=\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\cos \theta$
$$\begin{gathered} \Rightarrow \frac{9}{2}=\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\cos 60° \\ \Rightarrow \frac{9}{2}=\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\times \frac{1}{2} \\ \Rightarrow \left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|=9 \\ \Rightarrow \left|\overrightarrow{a}\right|\left|\overrightarrow{a}\right|=9\left(\mathrm{From}\left(\mathrm{i}\right)\right) \\ \Rightarrow {\left|\overrightarrow{a}\right|}^2=9 \\ \Rightarrow \left|\overrightarrow{a}\right|=\left|\overrightarrow{b}\right|=3 \end{gathered}$$