Question

Let $\vec{n}$ be a vector of magnitude $2\sqrt{3}$ such that it makes acute equal angles with the coordinate axes, the vector fo rm of equation of plane passing through $(1,-1,2)$ and normal to $\vec{n}$ is

• A. $\vec{r}\cdot \left(\hat{i}+2\hat{j}+3\hat{k}\right)=2$
• B. $\vec{r}\cdot \left(2i-3j+k\right)=2$
• C. $\vec{r}\cdot \left(2\hat{i}+\hat{j}+\hat{k}\right)=2$
• D. $\vec{r}\cdot \left(\hat{i}+\hat{j}+\hat{k}\right)=2$

Answer 1

Answer: (D)

Given $\alpha =\beta =\gamma$
$\therefore cos\alpha =cos\beta =cos\gamma$
$\Rightarrow l=m=n$
$\therefore l=\frac{1}{\sqrt{3}}$
$\left(\because l^2+m^2+n^2=1\right)<br>\therefore \vec{n}=2\sqrt{3}\left(\frac{\hat{i}+\hat{j}+\hat{k}}{\sqrt{3}}\right)$
$\left(\because \vec{r}=\left|\vec{r}\right|\left(l\hat{i}+m\hat{j}+n\hat{k}\right)\right)$
So, required vector equation of plane is
$\vec{r}\cdot \left(2\hat{i}+2\hat{j}+2\hat{k}\right)=\left(\hat{i}-\hat{j}+2\hat{k}\right)\cdot \left(2\hat{i}+2\hat{j}+2\hat{k}\right)$
or $\vec{r}\cdot \left(\hat{i}+\hat{j}+\hat{k}\right)=2$