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The plane — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsThe plane3 Marks
Question
The line $\vec{\text{r}}=\hat{\text{i}}+\lambda(2\hat{\text{i}}-\text{m}\hat{\text{j}}-3\hat{\text{k}})$ is parallel to the plane $\vec{\text{r}}\cdot(\text{m}\hat{\text{i}}+3\hat{\text{j}}+\hat{\text{k}})=4.$ Find m.
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Answer

The given line is parallel to the vector $\vec{\text{b}}=2\hat{\text{i}}-\text{m}\hat{\text{j}}-3\hat{\text{k}}$ and the given plane is normal to the vector $\vec{\text{n}}=\text{m}\hat{\text{i}}+3\hat{\text{j}}+\hat{\text{k}}$
If the line is parallel to the plane, the normal to the plane is perpendicular to the line.
$\Rightarrow\vec{\text{b}}\perp\vec{\text{n}}$
$\Rightarrow\vec{\text{b}}\cdot\vec{\text{n}}=0$
$\Rightarrow(2\hat{\text{i}}-\text{m}\hat{\text{j}}-3\hat{\text{k}})\cdot(\text{m}\hat{\text{i}}+3\hat{\text{j}}+\hat{\text{k}})=0$
$\Rightarrow2\text{m}-3\text{m}-3=0$
$\Rightarrow-\text{m}-3=0$
$\Rightarrow\text{m}=-3$

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