Question

$\mathrm{If}\left|\overrightarrow{a}\right|=8,\left|\overrightarrow{b}\right|=3\mathrm{and}\left|\overrightarrow{a}\times \overrightarrow{b}\right|=12,$ then the value of $\overrightarrow{a}·\overrightarrow{b}$ is

$$\begin{gathered} \left(\mathrm{a}\right)6\sqrt{3} \\ \left(\mathrm{b}\right)8\sqrt{3} \\ \left(\mathrm{c}\right)12\sqrt{3} \\ \left(\mathrm{d}\right)\mathrm{none}\mathrm{of}\mathrm{these} \end{gathered}$$

Answer 1

$$\begin{gathered} \mathrm{Given}\left|\overrightarrow{a}\right|=8 \\ \left|\overrightarrow{b}\right|=3 \\ \mathrm{and}\left|\overrightarrow{a}\times \overrightarrow{b}\right|=12 \\ \mathrm{using}\mathrm{Lagrange}\text{'}\mathrm{s}\mathrm{identity}, \\ {\left|\overrightarrow{a}\times \overrightarrow{b}\right|}^2={\left|a\right|}^2{\left|b\right|}^2-{\left(\overrightarrow{a}.\overrightarrow{b}\right)}^2 \\ \mathrm{i}.\mathrm{e}.{\left(12\right)}^2={\left(8\right)}^2{\left(3\right)}^2-{\left(\overrightarrow{a}.\overrightarrow{b}\right)}^2 \\ \mathrm{i}.\mathrm{e}.{\left(\overrightarrow{a}.\overrightarrow{b}\right)}^2=64\times 9-144 \\ \mathrm{i}.\mathrm{e}.{\left(\overrightarrow{a}.\overrightarrow{b}\right)}^2=432 \\ \mathrm{i}.\mathrm{e}.\overrightarrow{a}.\overrightarrow{b}=12\sqrt{3} \end{gathered}$$
Hence, the correct answer is option C.