Question

If $\hat{a}$ is any vector then ${(\hat{a}\times \hat{i})}^2+{(\hat{a}\times \hat{j})}^2+{(\hat{a}\times \hat{k})}^2=$

• A. $a^2$
• B. $2a^2$
• C. $3a^2$
• D. $4a^2$

Answer 1

The correct option is B $2a^2$

Let $\hat{a}=a_x\hat{i}+a_y\hat{j}+a_z\hat{k}$
Hence
$\hat{a}\times \hat{i}$$=-a_y\hat{k}+a_z\hat{j}$

Similarly
$\hat{a}\times \hat{j}$$=-a_x\hat{j}+a_z\hat{i}$
$\hat{a}\times \hat{k}$$=-a_x\hat{j}+a_y\hat{i}$

Taking the modulus of all the three and substituting in the equation, we get
$(\sqrt{a_x^2+a_y^2}{)}^2+(\sqrt{a_z^2+a_y^2}{)}^2+(\sqrt{a_x^2+a_z^2}{)}^2$
$=2[a_x^2+a_y^2+a_z^2]$
$=2|\overline{a}{|}^2$
$=2a^2$