how does one go about finding the Image of a vector P in a Plane x+y+z=0 {for example} ?
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.