For any vectora a the value of $(\vec{a} \times \hat{i})^2+(\vec{a} \times \hat{j})^2+(\vec{a} \times…
For any vectora a the value of $(\vec{a} \times \hat{i})^2+(\vec{a} \times \hat{j})^2+(\vec{a} \times \hat{k})^2$ is equal to
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$3 \vec{a}^2$
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$\vec{a}^2$
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$2 \vec{a}^2$
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$4 \vec{a}^2$
Solution
Let $\vec{a}=x \hat{i}+y \hat{j}+z \hat{k}$
$
\begin{aligned}
& \vec{a} \times \hat{\mathbf{i}}=z \hat{\mathbf{j}}-y \hat{\mathbf{k}} \\
& \Rightarrow(\vec{a} \times \hat{\mathbf{i}})^2=y^2+z^2
\end{aligned}
$
similarly $(\vec{a} \times \hat{j})^2=x^2+z^2$
and $(\vec{a} \times \hat{k})^2=x^2+y^2 \Rightarrow(\vec{a} \times \hat{i})^2=y^2+z^2$
similarly $(\vec{a} \times \hat{j})^2=x^2+z^2$
and $(\vec{a} \times \hat{k})^2=x^2+y^2$
$
\Rightarrow(\vec{a} \times \hat{i})^2+(\vec{a} \times \hat{j})^2+(\vec{a} \times \hat{k})^2=2\left(x^2+y^2+z^2\right)=2 \vec{a}^2 .
$
Asked in: JEE Main 2005
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