A) \[\vec{r}.(\hat{i}-\hat{j}+5\hat{k})=27\]
B) \[\vec{r}.(\hat{i}-\hat{j}+5\hat{k})=\sqrt{27}\]
C) \[\vec{r}.(5\hat{i}-\hat{j}+\hat{k})=\frac{1}{\sqrt{27}}\]
D) \[x-y-5z-27=0\]
Correct Answer: A
Solution :
Here, \[\vec{n}=\hat{i}-\hat{j}+5\hat{k}\] \[\hat{n}=\frac{{\vec{n}}}{|\vec{n}|}=\frac{\hat{i}-\hat{j}+5\hat{k}}{\sqrt{{{(1)}^{2}}+{{(-1)}^{2}}+{{(5)}^{2}}}}\] \[\frac{\hat{i}-\hat{j}+5\hat{k}}{\sqrt{27}}\] Also, length of perpendicular from origin to the plane is \[d=\sqrt{{{(1)}^{2}}+{{(-1)}^{2}}+{{(5)}^{2}}}=\sqrt{27}\] Now, equation of plane \[\vec{r}.\hat{n}=d\] \[\vec{r}.\,\frac{(\hat{i}-\hat{j}+5\hat{k})}{\sqrt{27}}=\sqrt{27}\] \[\Rightarrow \] \[\vec{r}.\,(\hat{i}-\hat{j}+5\hat{k})=27\]
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