A) \[\frac{1}{\sqrt{3}}\,(\hat{i}+\hat{j}-\hat{k})\]
B) \[\frac{1}{\sqrt{17}}\,(3\hat{i}+2\hat{j}-2\hat{k})\]
C) \[\frac{1}{\sqrt{13}}\,(3\hat{i}+2\hat{j}+2\hat{k})\]
D) \[\frac{1}{\sqrt{11}}\,(3\hat{i}+\hat{j}+\hat{k})\]
Correct Answer: B
Solution :
We have the given equation of plane \[3x+2y-2z=8\sqrt{17}\] \[\Rightarrow \] \[(x\hat{i}+y\hat{j}+z\hat{k}).(3\hat{i}+2\hat{j}-2\hat{k})=8\sqrt{17}\] Which is of the form \[\vec{r}.\,\,\vec{n}\,=\,d\] Where \[\vec{n}\] is the normal vector to the plane. Thus, required unit normal vector. \[\vec{n}=\frac{|\vec{n}|}{|\vec{n}|}=\frac{3\hat{i}+2\hat{j}-2\hat{k}}{\sqrt{9+4+4}}=\frac{1}{\sqrt{17}}\,(3\hat{i}+2\hat{j}-2\hat{k})\]
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