$$\textbf{Step 1 : Determining position vector }$$ $$\mathbf{\vec{a} }$$ $$\textbf{and normal vector}$$ $$\mathbf{\vec{n} }$$
$$\text{Let the plane passing through point P( 2 , 2 , 3).Given}$$
$$\text{the position vector of this point P be}$$
$$\vec{a} = 2 \hat{i} + 2 \hat{j} + 3 \hat{k}...................(i)$$
$$\text{and given that normal to plane having 3 , 4 , 3 as the direction ratios}$$
$$\text{then ,}$$
$$\Rightarrow \vec{n} = 3 \hat{i} + 4 \hat{j} + 3 \hat{k}..................(ii)$$
$$\textbf{Step 2 : Determining the vector equation of plane passing through the point }$$$$\mathbf{\vec{a} }$$
$$\textbf{and normal to the vector}$$ $$\mathbf{\vec{n} }$$
$$\text{the vector equation is }$$
$$( \vec{r} - \vec{a} )$$$${\cdot}$$ $$\vec{n} = 0$$
$$( \vec{r} - ( 2 \hat{i} + 2 \hat{j} + 3 \hat{k})) \cdot ( 3 \hat{i} + 4 \hat{j} + 3 \hat{k}) = 0.......................from \ (i)\ and \ (ii)$$
$$\textbf{Hence , the vector equation of plane passing is}$$ $$\mathbf{( \vec{r} - ( 2 \hat{i} + 2 \hat{j} + 3 \hat{k})) \cdot ( 3 \hat{i} + 4 \hat{j} + 3 \hat{k}) = 0}$$