If $\vec a =\hat i +\hat j, \vec b=2\hat j-\hat k$ and $\vec r×\vec a =\vec b ×\vec a, \vec r× \vec b=\vec a×\vec b$ then a unit vector in the direction of $\vec r$ is
Answer & explanation
Correct answer: option 1
We have,
$\vec r×\vec a =\vec b ×\vec a$ and $\vec r× \vec b=\vec a×\vec b$
$⇒\vec r×\vec a =-(\vec r× \vec b)$
$⇒\vec r×(\vec a +\vec b)=0$
$⇒\vec r$ is parallel to $\vec a +\vec b$
$⇒\vec r=λ(\vec a +\vec b)$
$⇒\vec r=λ(\hat i+3\hat j-\hat k)$
$⇒|\vec r|=\sqrt{11}λ$
∴ Required vector =$\frac{\vec r}{|\vec r|}=\frac{1}{\sqrt{11}}(\hat i+3\hat j-\hat k)$