Question

For any vector $x$, where $\hat{i},\hat{j},\hat{k}$ have their usual meanings the value of $(x\times \hat{i}{)}^2+(x\times \hat{j}{)}^2+(x\times \hat{k}{)}^2$ where $\hat{i},\hat{j},\mathbf{k}$ have their usual meanings, is equal to

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

Answer 1

Answer: (B)

Let $x=\alpha \hat{i}+\beta \hat{j}+\gamma k$
Then, $x\times \hat{i}=-\beta \hat{k}+\gamma \hat{j}$
$x\times \hat{j}=\hat{k}-\gamma \hat{i}$
$x\times k=-aj+\beta \hat{i}$
Now, $(x\times \hat{i}{)}^2=(x\times \hat{i})\cdot (x\times \hat{i})$
$=(-\beta \hat{k}+\gamma \hat{j})\cdot (-\beta \hat{k}+\gamma \hat{j})$
$={\beta }^2+{\gamma }^2$
Similarly, $(x\times \hat{j}{)}^2={\alpha }^2+{\gamma }^2$
and $(x\times \hat{K}{)}^2={\alpha }^2+{\beta }^2$
$\therefore (x\times \hat{i}{)}^2+(x\times \hat{j}{)}^2+(x\times \hat{K}{)}^2$
$={\beta }^2+{\gamma }^2+{\alpha }^2+{\gamma }^2+{\alpha }^2+{\beta }^2$
$=2\left({\alpha }^2+{\beta }^2+{\gamma }^2\right)=2|x{|}^2$