Question

If $\overline{a}$ is unit vector, then $|\overline{a}\times \hat{i}{|}^2+|\overline{a}\times \hat{j}{|}^2+|\overline{a}\times \hat{k}{|}^2=$ _____________.

• A. $2$
• B. $1$
• C. $0$
• D. $3$

Answer 1

The correct option is A $2$

Let $x,y$ and $z$ be the direction angles of the vector $\overrightarrow{a}$

Then,

$|\overrightarrow{a}\times \hat{i}|=|\overrightarrow{a}||\hat{i}|sin\left(x\right)=sinx$

Similarly,

$|\overrightarrow{a}\times \hat{j}|=siny$

$|\overrightarrow{a}\times \hat{k}|=sinz$

Therefore, $|\overrightarrow{a}\times \hat{i}{|}^2+|\overrightarrow{a}\times \hat{j}{|}^2+|\overrightarrow{a}\times \hat{k}{|}^2=si{n}^{2}x+si{n}^{2}y+si{n}^{2}z$

$=(1-co{s}^{2}x)+(1-co{s}^{2}y)+(1-co{s}^{2}z)$

$=3-(co{s}^{2}x+co{s}^{2}y+co{s}^{2}z)$

$=3-1$

$=2$

Hence, the answer is option (A).