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. $\vec{r}\cdot\left(2i-3 j+k\right)=2$
• 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 $\alpha = \beta = \gamma$
$\therefore cos\,\alpha = cos\,\beta = cos\,\gamma$
$\Rightarrow l=m=n$
$\therefore l=\frac{1}{\sqrt{3}}$
$\left(\because l^{2}+m^{2}+n^{2}=1\right)
\therefore \vec{n}=2\sqrt{3} \left(\frac{\hat{i}+\hat{j}+\hat{k}}{\sqrt{3}}\right)$
$\left(\because \vec{r}=\left|\vec{r}\right|\left(l \hat{i}+m \hat{j}+n \hat{k}\right)\right)$
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 $\vec{r}\cdot\left(\hat{i}+\hat{j}+\hat{k}\right)=2$