Question

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

Answer 1

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