For scalars lambda, mu,if the vector equation | vedclass

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(D) Given the vector equation of the plane is $r=(2+3 \lambda-\mu) \hat{i}+(1-2 \lambda+3 \mu) \hat{j}+(-2+2 \lambda+\mu) \hat{k}$ $\ldots$ $(i)$Let $

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$8 x+5 y-7 z-35=0$

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(D) Given the vector equation of the plane is $r=(2+3 \lambda-\mu) \hat{i}+(1-2 \lambda+3 \mu) \hat{j}+(-2+2 \lambda+\mu) \hat{k}$ $\ldots$ $(i)$Let $r=x \hat{i}+y \hat{j}+z \hat{k}$ $\ldots$ (ii)Comparing components from $(i)$ and (ii):$x = 2+3 \lambda-\mu$ $\ldots$ (iii)$y = 1-2 \lambda+3 \mu$ $\ldots$ (iv)$z = -2+2 \lambda+\mu$ $\ldots$ $(v)$Adding (iii) and $(v)$: $x+z = 2-2+3 \lambda+2 \lambda-\mu+\mu = 5 \lambda \Rightarrow \lambda = \frac{x+z}{5}$ $\ldots$ (vi)From $(v)$,$\mu = z+2-2 \lambda = z+2-2(\frac{x+z}{5}) = \frac{5z+10-2x-2z}{5} = \frac{-2x+3z+10}{5}$Substitute $\lambda$ and $\mu$ into (iv): $y = 1-2(\frac{x+z}{5})+3(\frac{-2x+3z+10}{5})$$5y = 5-2x-2z-6x+9z+30$$5y = -8x+7z+35$$8x+5y-7z-35=0$

For scalars $\lambda, \mu$,if the vector equation of a plane is $r=(2+3 \lambda-\mu) \hat{i}+(1-2 \lambda+3 \mu) \hat{j}+(-2+2 \lambda+\mu) \hat{k}$,then its Cartesian equation is

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