Question

Find the magnitude of two vectors $\overrightarrow{a}\mathrm{and}\overrightarrow{b}$that are of the same magnitude, are inclined at 60° and whose scalar product is 1/2.

Answer 1

$$\begin{gathered} \text{Given that the angle between}\overrightarrow{a}\text{and}\overrightarrow{b}{\text{is 30}}^0\text{.} \\ \text{Also,} \\ \left|\overrightarrow{a}\right|=\left|\overrightarrow{b}\right|;\overrightarrow{a}.\overrightarrow{b}=\frac{1}{2} \\ \text{We know that} \\ \overrightarrow{a}.\overrightarrow{b}=\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\cos \theta \\ \Rightarrow \frac{1}{2}=\left|\overrightarrow{a}\right|\left|\overrightarrow{a}\right|\cos 60 \\ \Rightarrow \frac{1}{2}={\left|\overrightarrow{a}\right|}^2\left(\frac{1}{2}\right) \\ \Rightarrow {\left|\overrightarrow{a}\right|}^2=1 \\ \Rightarrow \left|\overrightarrow{a}\right|=1 \\ \therefore \left|\overrightarrow{a}\right|=\left|\overrightarrow{b}\right|=1 \end{gathered}$$