If \( \hat{a} . \hat{b} \) and \( \hat{c} \) are the three unit vec...

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If \( \hat{a} . \hat{b} \) and \( \hat{c} \) are the three unit vectors and \( \alpha, \beta \) and \( \gamma \)
\( \mathrm{P} \) are scalars such that
\( \mathrm{W} \) \( \hat{c}=\alpha \hat{a}+\beta \hat{b}+\gamma(\hat{a} \times \hat{b}) \). If it is given that \( \hat{a} \cdot \hat{b}=0 \) and \( \hat{c} \) makes equal angle with both \( \hat{a} \) and \( \hat{b} \), then evaluate \( \alpha^{2}+\beta^{2}+\gamma^{2} \).
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