How to mirror vector A relative to a plane perpendicular to vector X?

For example in R3, vector A(a1,a2,a3) is mirrored to a plane perpendicular to vector X(x1,x2,x3). What is the vector resulted from this mirroring process ?|||Let's assume that ||X|| = 1 (if not, divide X by its norm; it doesn't alter the plane or A), then the component of A in the direction of X is:





A(X) = (A,X)X





Where (...,...) is the inner product. Notice that A(X), being collinear with X, is also orthogonal to the plane, and its mirror image is 鈭扐(X). Now, the mirror image of A in X is given by adding to A twice the mirror image of A(X):





M(A,X) = A 鈭?2A(X)





Where M(A,X) is the mirror image of A. This will be a lot clearer if you draw a picture.