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Question
Write the position vector of the point where the line \[\vec{r} = \vec{a} + \lambda \vec{b}\] meets the plane \[\vec{r} . \vec{n} = 0 .\]
Solution
\[\text{ Given equation of the line is } \]
\[ \vec{r} = a^\to + \lambda \vec{b} . . . \left( 1 \right)\]
\[\text{ Given equation of the plane is } \]
\[ \vec{r} . \vec{n} = 0\]
\[ \Rightarrow \left( \vec{a} + \lambda \vec{b} \right) . \vec{n} = 0................. [\text{ From } (1)]\]
\[ \Rightarrow \vec{a} . \vec{n} + \lambda \vec{b} . \vec{n} = 0\]
\[ \Rightarrow \lambda = - \left( \frac{\vec{a} . \vec{n}}{\vec{b} . \vec{n}} \right)\]
\[\text{ Substituting this in (1), we get} \]
\[ \vec{r} = \vec{a} - \left( \frac{\vec{a} . \vec{n}}{\vec{b} . \vec{n}} \right) \vec{b} , \text{ which is the required position vector that lies both on the line and the plane }.\]
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