If \(\vec{a} = 2\hat{i} + \hat{j} + 2\hat{k}\), then the value of \(|\hat{i} \times (\vec{a} \times \hat{i})|^2 + |\hat{j \times (\vec{a} \times \hat{j})|^2 + |\hat{k} \times (\vec{a} \times \hat{k})|^2\) is equal to:}
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Memorize this standard vector identity for objective tests: For any general vector \(\vec{a}\), the expansion identity \(\sum |\hat{i} \times (\vec{a} \times \hat{i})|^2\) always simplifies cleanly to \(2|\vec{a}|^2\).