The image of point $\displaystyle \mathrm{P}(x, y, z)$ with respect to line $\displaystyle \frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}$ is $\displaystyle \mathrm{P}^{\prime}(1,0,7)$. Find the coordinates of point P .

Let foot of the perpendicular on the given line from point P be $\displaystyle \mathrm{M}(\lambda, 2 \lambda+1,3 \lambda+2)$ D. ratios of $\displaystyle \mathbf{P P}^{\prime}$ are $\displaystyle \boldsymbol{\lambda}-\mathbf{1}, \mathbf{2} \boldsymbol{\lambda}+\mathbf{1}, \mathbf{3} \boldsymbol{\lambda}-\mathbf{5}$\begin{aligned} & 1(\lambda-1)+2(2 \lambda+1)+3(3 \lambda-5)=0 & \Rightarrow \lambda=1 \end{aligned}Coordinates of $\displaystyle \mathrm{M}(1,3,5)$\begin{aligned} & \frac{x+1}{2}=1, \frac{y+0}{2}=3, \frac{z+7}{2}=5 & \Rightarrow x=1, y=6, z=3 \Rightarrow \mathrm{P}(1,6,3) \end{aligned} $$