$2+$ $[$ Lines and Planes in Space (other).]

A plane given by the equation $x+2 y+3 z=0$ divides the space into two half-spaces. Determine whether the points ( $1,2,-2$ ) and ( $2,1,-1$ ) lie in the same or in different half-spaces.
How to define a half-space using an inequality

## Solution

It is not difficult to verify that one of the half-spaces (the one in the direction in which the normal vector (1, 2, 3) points) is defined by the inequality $x+2y+3z>0$, and the other half-space by the inequality $x+2y+3z<0$. Thus, they lie in different half-spaces.

## Answer

in different.