Question

$\hat{a}$ and $\hat{b}$ are mutually perpendicular unit vectors. If $\hat{r}$ is a vector satisfying $\hat{r}.\hat{a}=0,$ $\hat{r}.\hat{b}=1,$ and $[r$ $a$ $b]=1$, then $\hat{r}$ is

• A. $\hat{a}\times \hat{b}+\hat{b}$
• B. $\hat{a}+\hat{b}\times \hat{a}$
• C. $\hat{a}\times \hat{b}-\hat{b}$
• D. none of these

Answer 1

The correct option is A $\hat{a}\times \hat{b}+\hat{b}$

Let $r=x_{1}a+x_{2}b+x_3(a\times b)$

Then, $r.a=x_1{\left|a\right|}^2,r.b=x_2{\left|b\right|}^2$

and, $r.(a\times b)=x_3{|a\times b|}^2$

$\Rightarrow x_1=0,x_2=1,x_3=1$

$\therefore r=a\times b+b$