The value of $|a \times \hat{i}|^2+|a \times \hat{j}|^2+|a \times \hat{k}|^2$ is

$2 a^2$

$|\vec{a} \times \hat{i}|^2+|\vec{a} \times \hat{j}|^2+|\vec{a} \times \hat{k}|^2$

$\Rightarrow|a|^2 \sin ^2 \alpha+|a|^2 \sin ^2 \beta+|a|^2 \sin ^2 \gamma$

$=3\left|a^2\right|-\left|a^2\right|\left(\cos ^2 \alpha+\cos ^2 \beta+\cos ^2 \gamma\right)$

$=2\left|a^2\right|=2 a^2$

Hence (2) is correct answer.