Question

If $\overline{a}=2\overline{i}-2\overline{j}+\overline{k}$ then $\left[\begin{array}{ccc}\overline{i}& \overline{a}& \overline{a}\times \overline{i}\end{array}\right]+\left[\begin{array}{ccc}\overline{j}& \overline{a}& \overline{a}\times \overline{j}\end{array}\right]+\left[\begin{array}{ccc}\overline{k}& \overline{a}& \overline{a}\times \overline{k}\end{array}\right]=?$

• A. $9$
• B. $-9$
• C. $18$
• D. $8$

Answer 1

The correct option is D $8$

$\begin{array}{c}\overrightarrow{a}=2\hat{i}-2\hat{j}+\hat{k}\\ \left[\hat{i}.\overrightarrow{a}\overrightarrow{a}\times \overrightarrow{i}\right]+\left[\hat{j}\overrightarrow{a}\overrightarrow{a}\times \hat{j}\right]+\left[\hat{k}\overrightarrow{a}\overrightarrow{a}\times \hat{k}\right]=?\\ \left(\overrightarrow{a}\times \hat{i}\right).\left(\hat{i}\times \overrightarrow{a}\right)+\left(\overrightarrow{a}\times \hat{j}\right).\left(\hat{j}\times \overrightarrow{a}\right)+\left(\overrightarrow{a}\times \hat{k}\right).\left(\hat{k}\times \overrightarrow{a}\right)\\ \left[{\left|\overrightarrow{a}\times \hat{i}\right|}^2+{\left|\overrightarrow{a}\times \hat{j}\right|}^2+{\left|\overrightarrow{a}\times \hat{k}\right|}^2\right]\\ -\left[5+5+8\right]=-18\\ \overrightarrow{a}\times \hat{k}=\left(2\hat{i}-2\hat{j}+\hat{k}\right)\times \hat{k}\\ \overrightarrow{a}\times \hat{k}=-2\hat{j}-2\hat{i}\\ \left|\overrightarrow{a}\times \hat{k}\right|=8\\ Hence,\\ optionDiscorrectanswer.\end{array}$