Question:
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___________.
Solution:
$\hat{i} \times(\bar{a} \times \hat{i})=(\hat{i} \cdot \hat{i}) \bar{a}-(\hat{i} \cdot \bar{a}) \hat{i}=\hat{j}+2 \hat{k}$
Similarly, $\hat{j} \times(\bar{a} \times \hat{j})=2 \hat{i}+2 \hat{k}$
$\hat{k} \times(\bar{a} \times \hat{k})=2 \hat{i}+\hat{j}$
$\therefore|\hat{j}+2 \hat{k}|^{2}+|2 \hat{i}+2 \hat{k}|^{2}+|2 \hat{i}+\hat{j}|^{2}=5+8+5=18$
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