Plane is drawn from the point  and perpendicular to the line with direction ratios  to intersect the plane  . The locus of point Q is 

Locus of is the line of intersection of the plane Q  and 
$1(x-1)+1(y-1)+1(z-1)=0$

$$\begin{gathered} drs of the line=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& 2& 3\\ 1& 1& 1\end{array}\right|=\hat{i}-\hat{2}j+\hat{k} \\ \text{ Let the co-ordinate of Q be }(h,k,l) \\ \text{ Q lies on the plane }x+2y+3z=4\therefore \mathrm{h}+2\mathrm{k}+3\mathrm{l}=4....\left(1\right) \\ \begin{array}{r}(\mathrm{h}-1)\cdot 1+(\mathrm{k}-1)\cdot 1+(l-1)\cdot 1=0\\ \mathrm{h}+\mathrm{k}+l=3....2)\end{array} \\ \Rightarrow (h,k,l)=\left(0,5,-2\right) \\ \Rightarrow \text{ The line is }\frac{x}{1}=\frac{y-5}{-2}=\frac{z+2}{1} \end{gathered}$$