pls answer question 165

Dear student

The equation of any plane through the line of intersection of the given planes is [\vec{r} . (2 i - 3 j + 4 k) - 1] + λ [\vec{r} . (i - j) + 4] = 0 \vec{r} . [(2 + λ) i - (3 + λ) j + 4 k] = 1 - 4 λ . . . (1) If plane (1) is prependicular to \vec{r} . (2 i - j + k) + 8 = 0, then [(2 + λ) i - (3 + λ) j + 4 k] . (2 i - j + k) = 0 [Using \vec{n_{1}} . \vec{n_{2}} = 0] \Rightarrow 2 (2 + λ) + (3 + λ) + 4 = 0 \Rightarrow 3 λ + 11 = 0 \Rightarrow λ = - \frac{11}{3} Putting λ = \frac{- 11}{3} in (1), we obtain the equation of the required plane as \vec{r} . \{(2 - \frac{11}{3}) i - (3 - \frac{11}{3}) j + 4 k\} = 1 + \frac{44}{3} \Rightarrow \vec{r} . (- 5 i + 2 j + 12 k) = 47

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