Let $\vec u=\hat i+\hat j, \vec v=\hat i-\hat j$ and $\vec w=\hat i+2\hat j+3\hat k$. If $\hat n$ is a unit vector such that $\vec u.\hat n=0$ and $\vec v.\hat n=0$, then $|\vec w.\hat n|$ is equal to

3

We have,

 $\vec u.\hat n=0$ and $\vec v.\hat n=0$

$∴\hat n⊥\vec u$ and $\hat n⊥\vec v$$⇒\hat n=±\frac{\vec u×\vec v}{|\vec u×\vec v|}$

Now, $\vec u×\vec v=(\hat i+\hat j)×(\hat i-\hat j)=-2\hat k$

$∴\hat n±\hat k$

Hence, $|\vec w.\hat n|=|(\hat i+2\hat j+3\hat k).(±\hat k)|=3$