Find the shortest distance from the point (2,1,-2) to the plane x+y-z=1.

Question

Use lagranges multipliers.

Answer 1

The unit normal vector of of the given plane is n = (1,1,-1) / sqrt(3)

We can pick any point Q on the plane, say Q=(1,0,0)

The shortest distance from point P(2,1,-2) to the plane is

|P-Q|•n = (1,2,-2) • (1,1,-1)/Sqrt(3) = 5/Srt(3)