Question

For any vector
$\overrightarrow{a},thevalueof(\overrightarrow{a}\times \hat{i}{)}^2+(\overrightarrow{a}\times \hat{j}{)}^2+(\overrightarrow{a}\times \hat{k}{)}^{2}is(a\left){\overrightarrow{a}}^2\right(b\left)3\overrightarrow{a}\right(c\left)4{\overrightarrow{a}}^2\right(d)2{\overrightarrow{a}}^2$

Answer 1

(d) Let
$\overrightarrow{a}=x\hat{i}+y\hat{j}+z\hat{k}\therefore {\overrightarrow{a}}^2=x^2+y^2+z^2\therefore \overrightarrow{a}\times \hat{i}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ x& y& z\\ 1& 0& 0\end{array}\right|=\hat{i}\left[0\right]-\hat{j}[-z]+\hat{k}[-y]=z\hat{j}-y\hat{k}\therefore (\overrightarrow{a}\times \hat{i}{)}^2=(z\hat{j}-y\hat{k}\left)\right(z\hat{j}-y\hat{k})=y^2+z^{2}Similarly,(\overrightarrow{a}\times \hat{j}{)}^2=x^2+z^{2}and(\overrightarrow{a}\times \hat{k}{)}^2=x^2+y^2\therefore (\overrightarrow{a}\times \hat{i}{)}^2+(\overrightarrow{a}\times \hat{j}{)}^2+(\overrightarrow{a}\times \hat{k}{)}^2=y^2+z^2+x^2+z^2+x^2+y^2=2(x^2+y^2+z^2)=2{\overrightarrow{a}}^2$