Let $\bar{n}$ be a vector of magnitude $3 \sqrt{3}$ such that it makes equal acute angles with the…

Let $\bar{n}$ be a vector of magnitude $3 \sqrt{3}$ such that it makes equal acute angles with the co-ordinate axes. Then the vactor equation of a plane passing through $(1,-1,2)$ and normal to $\bar{n}$ is
  1. $\bar{r} \cdot(\hat{i}+\hat{j}+\hat{k})=3$
  2. $\bar{r} \cdot(3 \hat{i}+3 \hat{j}+3 \hat{k})=12$
  3. $\bar{r} \cdot(3 \hat{i}+3 \hat{j}+3 \hat{k})=1$
  4. $\bar{r} \cdot(3 \hat{i}+3 \hat{j}+3 \hat{k})=6$

Solution

$\begin{aligned} & \vec{r} \cdot \vec{n}=\vec{a} \cdot \vec{n} \\ & \Rightarrow \vec{r} \cdot(\hat{i}+\hat{j}+\hat{k})=(\hat{i}-\hat{j}+2 \widehat{k}) \cdot(\hat{i}+\hat{j}+\hat{k}) \\ & \Rightarrow \vec{r} \cdot(\hat{i}+\hat{j}+\hat{k})=2 \\ & \Rightarrow \vec{r} \cdot(3 \hat{i}+3 \hat{j}+3 \hat{k})=6\end{aligned}$

Asked in: MHT CET 2022 (10 Aug Shift 2)

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