Question

If $\left|\overrightarrow{a}\right|=10,\left|\overrightarrow{b}\right|=2\mathrm{and}\overrightarrow{a}·\overrightarrow{b}=12,$ then the value of $\left|\overrightarrow{a}\times \overrightarrow{b}\right|$ is
(a) 5
(b) 10
(c) 14
(d) 16

Answer 1

$$\begin{gathered} \mathrm{Given}\left|\overrightarrow{a}\right|=10,\left|\overrightarrow{b}\right|=2 \\ \overrightarrow{a}.\overrightarrow{b}=12 \end{gathered}$$

$\begin{array}{rcl}\mathrm{Since}{\left|\overrightarrow{a}\times \overrightarrow{b}\right|}^2& =& {\left|a\right|}^2{\left|b\right|}^2-{\left(\overrightarrow{a}.\overrightarrow{b}\right)}^2\mathrm{By}\mathrm{lagrange}\text{'}\mathrm{s}\mathrm{identity}\\ & =& {\left(10\right)}^2{\left(2\right)}^2-{\left(12\right)}^2\\ & =& 100\times 4-144\\ & =& 400-144\end{array}$
$$\begin{gathered} \mathrm{i}.\mathrm{e}{\left|\overrightarrow{a}\times \overrightarrow{b}\right|}^2=256 \\ \mathrm{i}.\mathrm{e}\left|\overrightarrow{a}\times \overrightarrow{b}\right|=16 \end{gathered}$$
Hence, the correct answer is option D.