How to calculate the mirror point along a line?

Question

In 2D plane, I have a point and a line. How to get the mirror point along this line?

Answer 1

I develop JS code which calculate formula from Ted Hop answer : Q' = Q + 2(I - nn^T)(P - Q) transformed to

Q' = Q + 2(I - d²vv^T)(P - Q)

Where column vector v=R-P is normal to line but not unit (like n) where P and R are two arbitrary line points; the vv^T is outer product; the d=1/sqrt(vx*vx + vy*vy) is inverse length of v. Because we use d² (=r in code) we can optimize calculations by omit sqrt

// 2D Points P=[x,y] and R are points on line, 
// Q is point for which we want to find reflection
function mirror(Q,[P,R]) {
  let [vx,vy]= [ R[0]-P[0], R[1]-P[1] ];
  let [x,y]  = [ P[0]-Q[0], P[1]-Q[1] ];
  let r= 1/(vx*vx+vy*vy);
  return [ Q[0] +2*(x -x*vx*vx*r -y*vx*vy*r), 
           Q[1] +2*(y -y*vy*vy*r -x*vx*vy*r)  ];
}

console.log( mirror([3,2], [[2,4],[4,5]]) )