Question

Let $A$ , $B$ and $C$ are unit vectors suppose $A.B=A.C=0$ and angle between $B$ and $C$ is $\frac{\pi }{6}$ then

• A. $A=\pm 2(B\times C)$
• B. $A=\pm \sqrt{2}(B\times C)$
• C. $A=\pm 3(B\times C)$
• D. $A=\pm \sqrt{3}(B\times C)$

Answer 1

The correct option is A $A=\pm 2(B\times C)$

Given $A,B,C$ are unit vectors and $A.B=A.C=0$

Since $A.B=0$ , $A$ is perpendicular to $B$

Since $A.C=0$ , $A$ is perpendicular to $C$

So $A$ is parallel to $B\times C$

$\Rightarrow A=\lambda (B\times C)$

Given that angle between $B$ and $C$ is ${30}^0$

Apply det on both sides for above equation , we get $|A|=|\lambda ||B||C|sin\left(30\right)$

$\Rightarrow |\lambda |=2$

$\Rightarrow \lambda =\pm 2$

So we get $A=\pm 2(B\times C)$

Therefore option $A$ is correct