If a=2 i+ j+2 k, then | i( a i)|^2+| j( a j)|^2+| k( a k)|^2 equals ________.

Question

Answer: 18

Accepted Answer

$\hat i\times(\vec a\times\hat i) = (\hat i\cdot\hat i)\vec a-(\hat i\cdot\vec a)\hat i = \vec a-2\hat i = \hat j+2\hat k$; similarly for $\hat j,\hat k$: $\hat j\times(\vec a\times\hat j)=2\hat i+2\hat k$, $\hat k\times(\vec a\times\hat k)=2\hat i+\hat j$.

$$|\hat j+2\hat k|^2+|2\hat i+2\hat k|^2+|2\hat i+\hat j|^2 = 5+8+5=18$$

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