Find the Cartesian equation of the plane \(\vec{r}.[(λ+2μ) \hat{i}+(2λ-μ) \hat{j}+(3λ-2μ)\hat{k}]\)=12.

(a) (λ-μ)x+y+(3λ-2μ)z=12

(b) (λ+3μ)x+(2+μ)y+(3λ-2μ)z=12

(c) (λ+2μ)x-2λy+(3λ-2μ)z=12

(d) (λ+2μ)x+(2λ-μ)y+(3λ-2μ)z=12

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My question is based upon Three Dimensional Geometry in section Three Dimensional Geometry of Mathematics – Class 12

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Correct choice is (d) (λ+2μ)x+(2λ-μ)y+(3λ-2μ)z=12

Best explanation: Given that the equation of the plane is \vec{r}.[(λ+2μ) \hat{i}+(2λ-μ) \hat{j}+(3λ-2μ)\hat{k}]=12

We know that, \vec{r}=x\hat{i}+y\hat{j}+z\hat{k}

∴(x\hat{i}+y\hat{j}+z\hat{k}).([(λ+2μ) \hat{i}+(2λ-μ) \hat{j}+(3λ-2μ)\hat{k}])=12

⇒(λ+2μ)x+(2λ-μ)y+(3λ-2μ)z=12 is the Cartesian equation of the plane.