Gujarat BoardEnglish MediumSTD 12 ScienceMathsThe plane3 Marks
Question
Find a unit normal vector to the plane x + 2y + 3z - 6 = 0
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Answer
The given equation of the plane is x + 2y + 3z - 6 = 0 x + 2y + 3z = 6 $\Rightarrow\vec{\text{r}}\cdot(\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}})=6$ Or $\vec{\text{r}}\cdot\vec{\text{n}}=6$ When, $\vec{\text{n}}=\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}}\ ...(\text{i})$ Now, $|\vec{\text{n}}|=\sqrt{1^2+2^2+3^2}$ $=\sqrt{1+4+9}$ $=\sqrt{14}$ Unit vector to the plane, $\hat{\text{n}}=\frac{\vec{\text{n}}}{|\vec{\text{n}}|}$ $=\frac{\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}}}{\sqrt{14}}$ $=\frac{1}{\sqrt{14}}\hat{\text{i}}+\frac{2}{\sqrt{14}}\hat{\text{j}}+\frac{3}{\sqrt{14}}\hat{\text{k}}$
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