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# For any vector $\overrightarrow a$ the value of $(\overrightarrow a\times \hat i)^2+(\overrightarrow a\times \hat j)^2+(\overrightarrow a\times \hat i)^2=?$

Can you answer this question?

Let $\overrightarrow a=x\hat i+y\hat j+z\hat k$
$(\overrightarrow a\times i)^2=(z\hat j-y\hat k)^2=z^2+y^2$
Similarly $(\overrightarrow a\times \hat j)^2=x^2+z^2\:\:and\:\:(\overrightarrow a\times \hat k)^2=x^2+y^2$
$\therefore\:(\overrightarrow a\times i)^2+(\overrightarrow a\times j)^2+(\overrightarrow a\times k)^2$=
$y^2+z^2+x^2+z^2+x^2+y^2=2(x^2+y^2+z^2)$
$=2|\overrightarrow a|^2$
answered Nov 10, 2013