The vector equation of the plane passing through the intersection of the planes $$\overrightarrow r .\left( {\widehat i + \widehat j + \widehat k} \right) = 1$$ and $$\overrightarrow r .\left( {\widehat i - 2\widehat j} \right) = - 2$$, and the point (1, 0, 2) is :
JEE · Math · previous-year question
- A.$$\overrightarrow r .\left( {\widehat i + 7\widehat j + 3\widehat k} \right) = {7 \over 3}$$
- B.$$\overrightarrow r .\left( {\widehat i + 7\widehat j + 3\widehat k} \right) = 7$$correct
- C.$$\overrightarrow r .\left( {3\widehat i + 7\widehat j + 3\widehat k} \right) = 7$$
- D.$$\overrightarrow r .\left( {\widehat i - 7\widehat j + 3\widehat k} \right) = {7 \over 3}$$
Answer
B. $$\overrightarrow r .\left( {\widehat i + 7\widehat j + 3\widehat k} \right) = 7$$
Explanation
Given, point (1, 0, 2) Equation of plane = $$\overrightarrow r\,.\,(\widehat i + \widehat j + \widehat k) = 1$$ and $$\overrightarrow r\,.\,(\widehat i - 2\widehat j) = - 2$$ Equation of plane passing through the intersection of given planes is $$[\overrightarrow r\,.\,(\widehat i + \widehat j + \widehat k) - 1] + \lambda [\overrightarrow r\,.\,(\widehat i - 2\widehat j) + 2] = 0$$ $$\because$$ This plane passes through point (1, 0, 2) i.e., vector $$(\widehat i + 2\widehat k)$$ $$\therefore$$ $$[(\widehat i + 2\widehat k)\,.\,(\widehat i + \widehat j + \widehat k) - 1] + \lambda [(\widehat i + 2\widehat k)\,.\,(\widehat i - 2\widehat j) + 2] = 0$$ $$ \Rightarrow (3 - 1) + \lambda (1 + 2) = 0$$ $$ \Rightarrow 2 + \lambda \times 3 = 0$$ $$ \Rightarrow \lambda = - 2/3$$ Hence, equation of required plane is $$[\overrightarrow r\,.\,(\widehat i + \widehat j + \widehat k) - 1] + \left( {{{ - 2} \over 3}} \right)[\overrightarrow r\,.\,(\widehat i - 2\widehat j) + 2] = 0$$ $$ \Rightarrow $$ $$3[\overrightarrow r\,.\,(\widehat i + \widehat j + \widehat k) - 1] - 2[\overrightarrow r\,.\,(\widehat i - 2\widehat j) + 2] = 0$$ $$ \Rightarrow $$ $$\overrightarrow r\,.\,(\widehat i + 7\widehat j + 3\widehat k) = 7$$
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