After three mirrors at right angles the beam leaves as e''' = −e. For your sliders e = (−0.58 i − 0.58 j − 0.58 k) comes out as e''' = (0.58 i + 0.58 j + 0.58 k), and the angle between them is 180°.
Problem statement
Show that a light beam reflected in succession from three mutually perpendicular plane mirrors reverses its direction.
We assume the three mirrors are flat and meet at one corner, like the inside corner of a box. This means the three normals can be taken as the x, y and z axes.
We assume the beam meets mirror 1, then mirror 2, then mirror 3. This means the order is x, then y, then z.
We assume the beam is a straight ray described by a unit vector e. This means we follow only its direction.
Symbols: e is the unit vector of the incoming beam. e', e'' and e''' are the unit vectors after mirror 1, 2 and 3. i, j and k are the unit vectors along the x, y and z axes. The numbers a, b and c are the sizes of the x, y and z parts of e: a = sin θ cos φ, b = sin θ sin φ, c = cos θ, where θ is the angle from the z axis and φ the turn from the x axis.
The big idea in plain words
A mirror flips only the part of the beam that points into it. Mirror 1 faces along x, so it flips the x part and leaves y and z alone. Mirror 2 flips the y part. Mirror 3 flips the z part. After the three mirrors every part has been flipped once, so the whole vector has changed sign. That is a reversed beam. The table in the drawing shows it: the boxed number is the part each mirror flips. The drawing shows the three mirrors as the corner of a room, with the beam inside it. Drag the picture to turn the corner, use the wheel to zoom, and use the Move view button to slide it, and the small direction picture on the right shows the same thing: e and e''' are arrows from one point, pointing opposite ways.
The solution in each frame
Lab frame
The mirrors are at rest. The full solution is in the next section.
Other frames
Moving mirrors would need a second frame. This problem has fixed mirrors, so one frame is enough.
The solution step by step
Choose the axes along the mirror normals. Write the incoming beam as
The reflection rule for a mirror with unit normal n is e' = e − 2 n (e · n). For mirror 1, n = i, so e · n = e_x:
Mirror 2 has n = j. The beam arriving at it is e', whose y part is e_y:
Mirror 3 has n = k:
So the beam leaves in the direction opposite to the one it arrived in. This is why a corner of three mirrors is used as a reflector on bicycles and road signs: the light goes back toward its source.
For your sliders e = (−0.58 i − 0.58 j − 0.58 k), e' = (0.58 i − 0.58 j − 0.58 k), e'' = (0.58 i + 0.58 j − 0.58 k) and e''' = (0.58 i + 0.58 j + 0.58 k).
The page applies e' = e − 2 n (e · n) three times with n = i, j, k and compares the result with −e. The two agree to better than 1e-12 for every slider value. The angle between e and e''' is always 180°.
Five methods
1. Force method
Each smooth mirror pushes the light only along its own normal. So each mirror changes only one part of the momentum, and reverses it. Three mirrors reverse all three parts.
2. Energy method
The mirrors do not change the energy, so the length of e stays 1. A sign change of every part keeps the length: a² + b² + c² = 0.577² + 0.577² + 0.578² = 1.
3. Graph method
The first graph follows each part of the beam through the mirrors: the x part flips after mirror 1, the y part after mirror 2, the z part after mirror 3. The second graph shows exit part against incoming part: a straight line with slope −1. The third graph shows the angle between e and e''' as a flat line at 180°, whatever a is.
4. Calculus method
Not needed. This is a result in vector algebra.
5. Check method
Limits: a beam along −z (a = b = 0) meets only the z flip and comes back along +z. A beam at 45° to two axes flips two parts and then the third. In every case e''' = −e.
Fastest: flip one part per mirror. Most reliable: the three-step formula e' = e − 2 n (e · n).
What each slider changes
| Control | Changes the answer? | What it changes |
|---|---|---|
| θ, angle of the beam from the vertical | Yes | a, b, c, the path, the three angles of incidence, the table, the direction picture and the graphs |
| φ, turn from the x axis | Yes | a, b, the path, the angles at the mirrors, the table, the direction picture and the graphs |
| Turn, tilt, zoom and move the view (drag, wheel, sliders, buttons) | No (view only) | Turns, tilts, enlarges and moves the corner of mirrors so the beam can be seen from any side while it keeps reflecting |
Parameters
| Quantity | Symbol | Value used |
|---|---|---|
| Angle from the vertical | θ | 54.7° |
| Turn from the x axis | φ | 45° |
| Angle of incidence at mirrors 1, 2, 3 | 54.8°, 54.8°, 54.7° | |
| x part | a | 0.577 |
| y part | b | 0.577 |
| z part | c | 0.578 |
| Angle between e and e''' | e to e''' | 180° |
| Angle between e and e' | e to e' | 70.49° |
Exam traps
- Flipping the wrong part: mirror 1 flips x, not y or z.
- Forgetting that e' is the beam that arrives at mirror 2, so its y part is the one to flip.
- Using mirrors that are not at exactly 90°. Then the beam returns at a small angle, not exactly back.
- Treating e as a position. It is a direction, so it has unit length.
- Thinking the order matters for the final direction. It does not, though it changes the path.