For any point $(x, y, z)$ on the plane, the distance from the origin to that point is:
$$
\sqrt{x^2+y^2+z^2}
$$
But most of those points are not the closest point - they are farther away because you're not approaching perpendicularly.
The closest point on the plane is the one where this distance is smallest.
We want the shortest distance from the origin $(0,0,0)$ to the plane
$$
x+y-2 z=6
$$
The shortest distance is the perpendicular distance, which is the minimum of
$$
\sqrt{x^2+y^2+z^2}
$$
for all points $(x, y, z)$ on the plane.
