Question

For any vector $\overrightarrow{x}$, the value of $|\overrightarrow{x}\times \hat{i}{|}^2+|\overrightarrow{x}\times \hat{j}{|}^2+|\overrightarrow{x}\times \hat{k}{|}^2$ is equal to

• A. $|\overrightarrow{x}{|}^2$
• B. $2|\overrightarrow{x}{|}^2$
• C. $3|\overrightarrow{x}{|}^2$
• D. $4|\overrightarrow{x}{|}^2$

Answer 1

The correct option is B $2|\overrightarrow{x}{|}^2$

Let $x=ai+bj+ck$
$x\times i=(ai+bj+ck)\times i=-bk+cj$
$x\times j=(ai+bj+ck)\times j=ak-ci$
$x\times k=(ai+bj+ck)\times k=-aj+bi$
$|x\times i{|}^2+|x\times j{|}^2+|x\times k{|}^2=b^2+c^2+a^2+c^2+a^2+b^2$
$|x\times i{|}^2+|x\times j{|}^2+|x\times k{|}^2=2(a^2+b^2+c^2)=2|x{|}^2$