Question

If $\overrightarrow{a}=\hat{i}-2\hat{j}+3\hat{k},\mathrm{and}\overrightarrow{b}=2\hat{i}+3\hat{j}-5\hat{k},$ then find $\overrightarrow{a}\times \overrightarrow{b}.$ Verify that $\overrightarrow{a}\mathrm{and}\overrightarrow{a}\times \overrightarrow{b}$are perpendicular to each other.

Answer 1

$$\begin{gathered} \mathrm{Given}: \\ \overrightarrow{a}=\stackrel{⏜}{i}-2\stackrel{⏜}{j}+3\stackrel{⏜}{k} \\ \overrightarrow{b}=2\stackrel{⏜}{i}+3\stackrel{⏜}{j}-5\stackrel{⏜}{k} \end{gathered}$$

$$\begin{gathered} \overrightarrow{a}\times \overrightarrow{b}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& -2& 3\\ 2& 3& -5\end{array}\right| \\ =\hat{i}+11\hat{j}+7\hat{k} \\ \text{Now,} \\ \overrightarrow{a}.\left(\overrightarrow{a}\times \overrightarrow{b}\right)=1-22+21 \\ =0 \\ \mathrm{Thus},\overrightarrow{a}\text{is perpendicular to}\overrightarrow{a}\times \overrightarrow{b}. \end{gathered}$$