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Line and Plane — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsLine and Plane1 Mark
Question
Find the vector equation of the plane passing through the point having position vector $2 \hat{i}+3 \hat{j}+4 \hat{k}$ and perpendicular to the vector $2 \hat{i}+\hat{j}-2 \hat{k}$.
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Answer

We know that the vector equation of the plane passing through $\mathrm{A}(\bar{a})$ and normal to vector $\bar{n}$ is given by $\bar{r} \cdot \bar{n}=\bar{a} \cdot \bar{n}$.
Here $\bar{a}=2 \hat{i}+3 \hat{j}+4 \hat{k}, \bar{n}=2 \hat{i}+\hat{j}-2 \hat{k}$
$
\bar{a} \cdot \bar{n}=(2 \hat{i}+3 \hat{j}+4 \hat{k}) \cdot(2 \hat{i}+\hat{j}-2 \hat{k})=4+3-8=-1
$
The vector equation of the plane is $\bar{r} \cdot(2 \hat{i}+\hat{j}-2 \hat{k})=-1$.

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