Question

If the angle between the planes $\overline{r}.(m\hat{i}-\hat{j}+2\hat{k})+3=0$ and $\overline{r}.(2\hat{i}-m\hat{j}-\hat{k})-5=0$ is $\frac{\pi }{3}$ then $m=$

• A. $2$
• B. $\pm 3$
• C. $3$
• D. $-2$

Answer 1

The correct option is C $3$

Direction ratios $\overline{n_1}$ are $m,-1,2$
Direction ratios $\overline{n_2}$ are $2,-m,-1$

$\theta =\frac{\pi }{3}$
$\cos \theta =\left|\frac{\overline{n_1}.\overline{n_2}}{|\overline{n_1}||\overline{n_2}|}\right|\Rightarrow \frac{1}{2}=\left|\frac{2m+m-2}{\sqrt{m^2+5}\sqrt{m^2+5}}\right|$
$\frac{1}{2}=\left|\frac{3m-2}{m^2+5}\right|\Rightarrow m^2+5=\pm (6m-4)$

$\Rightarrow m^2+5=6m-4,m^2+5=-6m+4$
$m^2-6m+9=0,m^2+6m+1=0$

$(m-3{)}^2=0$

$m=3$