Question

The perpendicular distance from origin to the plane passing through the points $2\overline{i}+\overline{j}+3\overline{k},\overline{i}+3\overline{j}+2\overline{k},3\overline{i}+2\overline{j}+\overline{k}$ is

• A. $\sqrt{3}$
• B. $2\sqrt{3}$
• C. $\frac{\sqrt{3}}{2}$
• D. $\sqrt{\frac{3}{2}}$

Answer 1

The correct option is B $2\sqrt{3}$

$\begin{array}{c}\left(r-a\right).\left(\left(b-a\right)\times \left(c-a\right)\right)=0\\ \left\{r-\left(2\hat{i}+\hat{j}+3\hat{k}\right)\right\}.\left\{\left(-\hat{i}+2\hat{j}-\hat{k}\right)\times \left(\hat{i}+\hat{j}-2\hat{k}\right)\right\}=0\\ \left\{r-\left(2\hat{i}+\hat{j}+3\hat{k}\right)\right\}.\left(-3\hat{i}-3\hat{j}-3\hat{k}\right)=0\\ \left(-3x-3y-3z\right)+\left(6+3+9\right)=0\\ -3x-3y-3z+18=0\\ -x-y-z+6=0\\ equationofplanex+y+z-6=0\\ perpendiculardis\tan cefromorigintotheplane\\ =\frac{\left|-6\right|}{\sqrt{1+1+1}}\\ =\frac{6}{\sqrt{3}}\\ =2\sqrt{3}unit\\ Hence,\\ optionBiscorrectanswer.\end{array}$