Question

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

• A. $2$
• B. $4$
• C. $1$
• D. $0$

Answer 1

Answer: (A)

$2$

Let $\overrightarrow{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$

Putting it in the equation,

$|(a_1\hat{i}+a_2\hat{j}+a_3\hat{k})\times \hat{i}{|}^2$ + $|(a_1\hat{i}+a_2\hat{j}+a_3\hat{k})\times j{|}^2$ + $|(a_1\hat{i}+a_2\hat{j}+a_3\hat{k})\times \hat{k}{|}^2$

=> $|a_3\hat{i}-a_2\hat{k}{|}^2$+$|a_1\hat{k}-a_3\hat{i}{|}^2$+$|a_2\hat{i}-a_1\hat{j}{|}^2$

=> $2(a_3{)}^2+2(a_1{)}^2+2(a_2{)}^2$

=>$2|\overrightarrow{a}{|}^2=2$ (as $|\overrightarrow{a}|=1$)