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The plane — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsThe plane2 Marks
Question
Write the position vector of the point where the line $\vec{\text{r}}=\vec{\text{a}}+\lambda\vec{\text{b}}$ meets the plane $\vec{\text{r}}.\vec{\text{n}}=0$.
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Answer

The given equation of the plane is
$\vec{\text{r}}=\vec{\text{a}}+\lambda\vec{\text{b}}\ ....(1)$
Given equation of the plane is
$\vec{\text{r}}.\vec{\text{n}}=0$
$\Rightarrow\big(\vec{\text{a}}+\lambda\vec{\text{b}}\big)\vec{\text{n}}=0$ [ From (1) ]
$\Rightarrow\vec{\text{a}}.\vec{\text{n}}+\lambda\vec{\text{b}}.\vec{\text{n}}=0$
$\Rightarrow\lambda=-\Big(\frac{\vec{\text{a}}.\vec{\text{n}}}{\vec{\text{b}}.\vec{\text{n}}}\Big)$
Substituting this in (1), we get
$\vec{\text{r}}=\vec{\text{a}}-\Big(\frac{\vec{\text{a}}.\vec{\text{n}}}{\vec{\text{b}}.\vec{\text{n}}}\Big)\vec{\text{b}},$ which is the required position vector that lies both on the line and the plane.

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