Let \( P \) be a plane passing through the points \( (1,0,1),(1,-2,...

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Let \( P \) be a plane passing through the points \( (1,0,1),(1,-2,1) \) and \( (0,1,-2) \). Let a vector
P \( \vec{a}=\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k} \) be such that \( \vec{a} \) is parallel to the plane \( \mathrm{P} \), perpendicular to \( (\hat{i}+2 \hat{j}+3 \hat{k}) \) and \( \vec{a} \cdot(\hat{i}+\hat{j}+2 \hat{k})=2 \), then \( (\alpha-\beta+\gamma)^{2} \) equals
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