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 } $
$x \times k =- a j +\beta \hat{ i }$
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 $(x \times \hat{K} )^{2}=\alpha^{2}+\beta^{2}$
$\therefore (x \times \hat{i})^{2}+(x \times \hat{j})^{2}+(x \times \hat{K} )^{2}$
$=\beta^{2}+\gamma^{2}+\alpha^{2}+\gamma^{2}+\alpha^{2}+\beta^{2}$
$=2\left(\alpha^{2}+\beta^{2}+\gamma^{2}\right)=2|x|^{2}$