Question

(i) If $\overrightarrow{a}=3\hat{i}+4\hat{j}\mathrm{and}\overrightarrow{b}=\hat{i}+\hat{j}+\hat{k},$ find the value of $\left|\overrightarrow{a}\times \overrightarrow{b}\right|.$

(ii) If $\overrightarrow{a}=2\hat{i}+\hat{k},\overrightarrow{b}=\hat{i}+\hat{j}+\hat{k},$ find the magnitude of $\overrightarrow{a}\times \overrightarrow{b}.$

Answer 1

$$\begin{gathered} \left(\mathrm{i}\right)\mathrm{Given}: \\ \overrightarrow{a}=3\hat{i}+4\hat{j}+0\hat{k} \\ \overrightarrow{b}=\hat{i}+\hat{j}+\hat{k} \\ \therefore \overrightarrow{a}\times \overrightarrow{b}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 3& 4& 0\\ 1& 1& 1\end{array}\right| \\ =\left(4-0\right)\hat{i}-\left(3-0\right)\hat{j}+\left(3-4\right)\hat{k} \\ =4\hat{i}-3\hat{j}-\hat{k} \\ \Rightarrow \left|\overrightarrow{a}\times \overrightarrow{b}\right|=\sqrt{16+9+1} \\ =\sqrt{26} \end{gathered}$$

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