# Solution using complex numbers

A ray of light is travelling along $\mathbf{i}+\sqrt{3} \, \mathbf{j}$, it hits a plane mirror and is reflected along $\mathbf{i}-\sqrt{3} \, \mathbf{j}$. What is the angle between normal and the incident wave? It looks like physics but only concept is angle of incidence = angle of reflection along the normal perpendicular to the mirror. I know the answer is $30^{\circ}$. But I am interested if there is a way using complex numbers. Like by calculating the argument by fixing mirror at the origin. Thanks!

• shouldn't that angle be just half of the angle between the incident and reflected ray. The latter can be computed using dot product of the two vectors. – Anurag A Apr 23 '16 at 16:47
• I can't think of a way or a reason to use complex numbers for such a problem, the simpler vector analysis is much easier – Triatticus Apr 23 '16 at 16:51
• i thought symmetry might help – Archis Welankar Apr 23 '16 at 17:15
• What motivated your question originally? I am always interested in the ways that the complex plane is isomorphic and yet also different from the real plane – Chill2Macht Apr 23 '16 at 19:50
• Ya like using Argand plane @ William krinsman – Archis Welankar Apr 24 '16 at 3:01