Question

If $\left|\overrightarrow{a}\times \overrightarrow{b}\right|=4,\left|\overrightarrow{a}·\overrightarrow{b}\right|=2,\mathrm{then}{\left|\overrightarrow{a}\right|}^2{\left|\overrightarrow{b}\right|}^2=$

(a) 6
(b) 2
(c) 20
(d) 8

Answer 1

(c) 20

$$\begin{gathered} \text{We know} \\ {\left(\overrightarrow{a}.\overrightarrow{b}\right)}^2+{\left|\overrightarrow{a}\times \overrightarrow{b}\right|}^2={\left|\overrightarrow{a}\right|}^2{\left|\overrightarrow{b}\right|}^2...\left(1\right) \\ \left|\overrightarrow{a}.\overrightarrow{b}\right|=2\left(\mathrm{Given}\right) \\ \Rightarrow {\left|\overrightarrow{a}.\overrightarrow{b}\right|}^2={\left(\overrightarrow{a}.\overrightarrow{b}\right)}^2 \\ \mathrm{From}\left(1\right),\mathrm{we}\mathrm{get} \\ {\left(2\right)}^2+{\left(4\right)}^2={\left|\overrightarrow{a}\right|}^2{\left|\overrightarrow{b}\right|}^2 \\ \Rightarrow {\left|\overrightarrow{a}\right|}^2{\left|\overrightarrow{b}\right|}^2=20 \end{gathered}$$