A laser is fired at a flat mirror.

The laser beam starts at the point (0,5) following the equation y = 5 - x/3 heading down to the right.
The mirror follows the equation y=x/2.

Find the equation for the path the beam takes after it hits the mirror.

The point of reflection is the intersection of lines

         y = 5 - x/3   and   y = x/2

                      or

                     (6,3)

The incident vector (towards source) is

      I = -3i + j

The normal vector is

      N = -i + 2j

The reflection vector is

      R = i + mj

Therefore,

       dot(I,N)     dot(R,N)
      ---------- = ----------
          |I|          |R|

                 or

       dot(-3i + j,-i + 2j)     dot(i + mj,-i + 2j)
      ---------------------- = ---------------------   
         sqrt(3^2 + 1^1)          sqrt(1^2 + m^2)

                 or

          5            -1 + 2m
      ---------- = ---------------
       sqrt(10)     sqrt(1 + m^2)

                 or

      (3m + 1)(m - 3) = 0

      m = -1/3 is the incident vector and

      m = 3 is the reflection vector

Therefore, the line y = mx + k through the point (6,3) is

      y = 3(x - 5)