n=1/sqrt(3)(1,1,1) is the unit normal vector of your plane.

Point of the plane are precisely the points fulfilling

X.n=0

(By dot I denote the scalar product.)

Let us denote P.n=r. Note that the point

C= P-rn

belongs to the plane. It is the projection of the point P to the plane. (Since we changed P in the direction perpendicular to the plane.)

C is in the middle between P and its reflection, so the reflection is

P'= P-2rn.