Question

If $\overline{a}=\overline{2i}-\overline{2j}+\overline{k}then[\overline{i}\overline{a}\overline{a}\times \overline{i}]+[\overline{j}\overline{a}\overline{a}\times \overline{j}]+[\overline{k}\overline{a}\overline{a}\times \overline{k}]=$

• A. - 9
• B. 9
• C. 18
• D. - 18

Answer 1

The correct option is D - 18

$-\{{(\overline{a}\times \overline{i})}^2+(\overline{a}\times \overline{j}{)}^2+(\overline{a}\times \overline{k}{)}^2\}=-2|\overline{a}{|}^2-\{{(\overline{a}\times \overline{i})}^2+(\overline{a}\times \overline{j}{)}^2+(\overline{a}\times \overline{k}{)}^2\}=-2|\overline{a}{|}^2[\hat{i}\overrightarrow{a}\overrightarrow{a}\times \hat{i}]+[\hat{j}\overrightarrow{a}\overrightarrow{a}\times \hat{i}]+[\hat{k}\overrightarrow{a}\overrightarrow{a}\times \hat{k}]=\hat{i}.(\overrightarrow{a}\times (\overrightarrow{a}\times \hat{i}\left)\right)+\hat{j}.(\overrightarrow{a}\times (\overrightarrow{a}\times \hat{j}\left)\right)+\hat{k}.(\overrightarrow{a}\times (\overrightarrow{a}\times \hat{k}\left)\right)=(\overrightarrow{a}\times \hat{i}).(\hat{i}\times \overrightarrow{a})+(\overrightarrow{a}\times \hat{j}).(\hat{j}\times \overrightarrow{a})+(\overrightarrow{a}\times \hat{k}).(\hat{k}\times \overrightarrow{a})=-\left[\right(\overrightarrow{a}\times \hat{i}{)}^2+(\overrightarrow{a}\times \hat{j}{)}^2+(\overrightarrow{a}\times \hat{k}{)}^2]=-[|\overrightarrow{a}{|}^2|\hat{i}{|}^{2}si{n}^2\alpha +|\overrightarrow{a}{|}^2|\hat{j}{|}^{2}si{n}^2\beta +|\overrightarrow{a}{|}^2|\hat{k}{|}^{2}si{n}^2\gamma ]=-|\overrightarrow{a}{|}^2[si{n}^2\alpha +si{n}^2\beta +si{n}^2\gamma ]=-|\overrightarrow{a}{|}^2[3-(co{s}^2\alpha +co{s}^2\beta +co{s}^2\gamma )]=-2|\overrightarrow{a}{|}^2=-2(4+4+1)=-18$