Question

$\mathrm{If}\overrightarrow{a}=4\hat{i}+3\hat{j}+\hat{k}\mathrm{and}\overrightarrow{b}=\hat{i}-2\hat{k},\mathrm{then}\mathrm{find}\left|2\hat{b}\times \overrightarrow{a}\right|.$

Answer 1

$$\begin{gathered} \mathrm{Given}: \\ \overrightarrow{a}=4\hat{i}+3\hat{j}+\hat{k} \\ 2\overrightarrow{b}=2\hat{i}+0\hat{j}-4\hat{k} \\ 2\overrightarrow{b}\times \overrightarrow{a}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 2& 0& -4\\ 4& 3& 1\end{array}\right| \\ =\left(0+12\right)\hat{i}-\left(2+16\right)\hat{j}+\left(6-0\right)\hat{k} \\ =12\hat{i}-18\hat{j}+6\hat{k} \\ \Rightarrow \left|2\overrightarrow{b}\times \overrightarrow{a}\right|=\sqrt{{12}^2+\left(-{18}^2\right)+6^2} \\ =\sqrt{504} \end{gathered}$$