I don't even know where to start on this one.
"Find the shortest distance from the point (1,0,-2) to the plane x+2y+z=4"
The equation of the plane can be written as...
$\displaystyle z(x,y)= 4-x-2 y$ (1)
... so that the squared distance from the point (1,0,-2) is...
$\displaystyle \delta^{2} (x,y)= (x-1)^{2}+ y^{2} + \{z(x,y)+2\}^{2}$ (2)
Now You minimize $\displaystyle \delta^{2}(x,y)$ respect to x and y and the problem is solved...
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$