Question

Write the value of $\left[\hat{i}+\hat{j}\hat{j}+\hat{k}\hat{k}+\hat{i}\right].$

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have} \\ \left[\hat{i}+\hat{j}\hat{j}+\hat{k}\hat{k}+\hat{i}\right]=\left[\left(\hat{i}+\hat{j}\right)\times \left(\hat{j}+\hat{k}\right)\right]·\left(\hat{k}+\hat{i}\right)\left(\because \left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]=\left(\overrightarrow{a}\times \overrightarrow{b}\right).\overrightarrow{c}\right) \\ =\left[\hat{i}\times \hat{j}+\hat{i}\times \hat{k}+\hat{j}\times \hat{j}+\hat{j}\times \hat{k}\right]·\left(\hat{k}+\hat{i}\right) \\ =\left[\hat{k}-\hat{j}+\hat{i}\right]·\left(\hat{k}+\hat{i}\right) \\ =\left[\left(\hat{k}·\hat{k}\right)+\left(\hat{k}·\hat{i}\right)-\left(\hat{j}·\hat{k}\right)-\left(\hat{j}·\hat{i}\right)+\left(\hat{i}·\hat{k}\right)+\left(\hat{i}·\hat{i}\right)\right] \\ =1+0-0-0+0+1=2 \end{gathered}$$