Mathematics - Vector Question with Solution | TestHub

Let $\vec{a}=x\hat{i}+y\hat{j}+z\hat{k}$ $$\begin{array}{rl}& \vec{a}\times \hat{\mathbf{i}}=z\hat{\mathbf{j}}-y\hat{\mathbf{k}}\\ & \Rightarrow (\vec{a}\times \hat{\mathbf{i}}{)}^2=y^2+z^2\end{array}$$ 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 $(\vec{a}\times \hat{k}{)}^2=x^2+y^2$ $$\Rightarrow (\vec{a}\times \hat{i}{)}^2+(\vec{a}\times \hat{j}{)}^2+(\vec{a}\times \hat{k}{)}^2=2\left(x^2+y^2+z^2\right)=2{\vec{a}}^2.$$