The vector $\vec r$ satisfying the conditions that

(i) it is perpendicular to $3\hat i+2\hat j+2\hat k$ and $18\hat i -22\hat j-5\hat k$

(ii) it makes an obtuse angle with y-axis,

(iii) $|\vec r|=14$, is

$2(-2\hat i-3\hat j+6\hat k)$

Let $\vec a = 3\hat i+2\hat j+2\hat k$ and $\vec b = 18\hat i -22\hat j-5\hat k$

Then, the required vector $\vec r$ is given by

$\vec r=λ(\vec a×\vec b)$

$⇒\vec r=λ(34\hat i+51\hat j-102\hat k)=17λ (2\hat i+3\hat j-6\hat k)$

Now, $|\vec r|=14⇒ 119|λ|=14⇒|λ|=\frac{2}{17}$

Since $\vec r$ makes an obtuse angle with y-axis. Therefore, $λ=-2/17$.

Hence, $\vec r=2(-2\hat i-3\hat j+6\hat k)$