Step 1
The required plane contains the line $ \large\frac{x-2}{2}=\large\frac{y-2}{3}=\large\frac{z-1}{3}$
$ \therefore $ it contains the point $A(p, v \: \overrightarrow a=2\overrightarrow i+2\overrightarrow j+\overrightarrow k)$ that lies on the line and it is parallel to the vector $ \overrightarrow u=2\overrightarrow i+3\overrightarrow j+3\overrightarrow k$
The plane is also parallel to the line $ \large\frac{x+1}{3}=\large\frac{y-1}{2}=\large\frac{z+1}{1}$ ans so it is parallel to $ \overrightarrow v=3\overrightarrow i+2\overrightarrow j+\overrightarrow k$
Step 2
The vector equation of the plane is of the form
$ \overrightarrow r=\overrightarrow a+s\overrightarrow u+t\overrightarrow v$
$\overrightarrow r=2\overrightarrow i+2\overrightarrow j+\overrightarrow k+s(2\overrightarrow i+2\overrightarrow j+3\overrightarrow k)+t(3\overrightarrow i+2\overrightarrow j+\overrightarrow k)$
Step 3
The cartesian equation is
$ \begin{vmatrix} x-x_1 & y-y_1 & z-z_1 \\ l_1 & m_1 & n_1 \\ l_2 & m_2 & n_2 \end{vmatrix} = 0$
$ (x_1, y_1, z_1) = (2, 2, 1), (l_1, m_1, n_1)=(2, 3, 3), (l_2, m_2, n_2) = (3, 2, 1)$
$ \therefore \begin{vmatrix} x-2 & y-2 & z-1 \\ 2 & 3 & 3 \\ 3 & 2 & 1 \end{vmatrix}=0$
$(x-2)(3-6)-(y-2)(2-9)+(z-1)(4-9)=0$
$ -3x+6+7y-14-5z+5=0$
$ -3x+7y-5z-3=0$
or $ 3x-7y+5z=-3$