Question

Evaluate the following:
(i) $\left[\hat{i}\hat{j}\hat{k}\right]+\left[\hat{j}\hat{k}\hat{i}\right]+\left[\hat{k}\hat{i}\hat{j}\right]$

(ii) $\left[2\hat{i}\hat{j}\hat{k}\right]+\left[\hat{i}\hat{k}\hat{j}\right]+\left[\hat{k}\hat{j}2\hat{i}\right]$

Answer 1

$$\begin{gathered} \left(\mathrm{i}\right)\mathrm{We}\mathrm{have} \\ \left[\hat{i}\hat{j}\hat{k}\right]+\left[\hat{j}\hat{k}\hat{i}\right]+\left[\hat{k}\hat{i}\hat{j}\right] \\ =\left[\hat{i}\hat{j}\hat{k}\right]+\left[\hat{i}\hat{j}\hat{k}\right]+\left[\hat{i}\hat{j}\hat{k}\right]\left(\because \left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]=\left[\overrightarrow{b}\overrightarrow{c}\overrightarrow{a}\right]=\left[\overrightarrow{c}\overrightarrow{a}\overrightarrow{b}\right]\right) \\ =3\left[\hat{i}\hat{j}\hat{k}\right] \\ =3\left(\hat{i}\times \hat{j}\right).\hat{k}\left(\because \left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]=\left(\overrightarrow{a}\times \overrightarrow{b}\right).\overrightarrow{c}\right) \\ =3\left(\hat{k}.\hat{k}\right) \\ =3\left(1\right)=3 \end{gathered}$$

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