Images in plane mirrors
Every time you check your reflection before leaving the house, you are watching the laws of reflection at work. A plane mirror is nothing more than a flat, polished surface, yet the image it throws back has a precise, predictable set of properties — same size, same way up, but strangely reversed left to right. Nothing about this is magic: every property of the image follows directly from one rule, the angle of incidence equals the angle of reflection, applied ray by ray to the light leaving the object. This section pins down exactly what that image looks like, proves geometrically why it sits as far behind the mirror as the object stands in front, and extends the idea to two facing mirrors, which multiply a single object into a receding corridor of images.
Because every ray from an object obeys at a plane mirror, the reflected rays only appear to diverge from a point behind the mirror at the same perpendicular distance as the object — giving a virtual, erect, same-size, laterally inverted image.
The core idea
A plane mirror does not bend or absorb the geometry of light; it simply reflects each incident ray so that the angle of incidence equals the angle of reflection, measured from the normal at the point of striking:
where: is the angle of incidence and is the angle of reflection, both measured from the normal (the perpendicular to the mirror at the point of reflection), and both lying in the same plane as the incident ray. Every characteristic of a mirror image is just this one rule, applied to a whole bundle of rays leaving the same point on the object.
Characteristics of the image formed by a single plane mirror. The image is virtual: the reflected rays only appear to come from a point behind the mirror, so nothing real is present there and the image cannot be caught on a screen. It is erect, standing the same way up as the object, never upside down. It is the same size as the object, so the magnification is exactly . The image distance behind the mirror equals the object distance in front of it — proved below. And it shows lateral inversion: left and right are interchanged, which is why text held up to a mirror reads backwards and your right hand looks like the image's left hand.
Two plane mirrors facing each other. Stand an object between two parallel plane mirrors and each mirror reflects not just the object but also the image already formed in the other mirror, so mirror one images the object, mirror two images that image, mirror one images that, and so on. Geometrically this produces an infinite series of images, receding into the distance. In practice only a limited number stay visible, because no real mirror reflects all the light hitting it — each successive image loses a little more brightness, exactly what you notice in a barber's shop or a two-mirror dressing room.
Everyday uses. Plane mirrors serve as looking glasses for grooming, as the flat reflecting elements inside a periscope, in shop and room design to make a space feel larger, and in optical instruments wherever a beam of light must be redirected without changing its size or shape.
The goal is to show, using nothing beyond the law of reflection and simple triangle geometry, that the perpendicular distance of the virtual image behind a plane mirror is always exactly equal to the perpendicular distance of the object in front of it, for any position of the object and any ray chosen from it.
Step 1. Set up the object and the normal ray. Let be a point object, and let be the foot of the perpendicular dropped from onto the mirror, so is the normal distance . A ray from travelling exactly along this normal strikes the mirror at and, since its angle of incidence is zero, reflects straight back along the same line.
Step 2. Take a second ray from at some angle. Let a second ray leave and strike the mirror at a different point , making angle of incidence with the normal at . By the law of reflection this ray leaves at angle on the other side of that normal.
Step 3. Extend both reflected rays backward. The first reflected ray, travelling back along extended, lies on the straight line through perpendicular to the mirror. The second reflected ray, extended backward through the mirror, meets this same perpendicular line at a point . Because both extensions are straight-line continuations of real reflected rays, is the point from which an observer's eye perceives the light to be diverging — the image.
Step 4. Compare triangles and . Both share side and both have a right angle at , since and lie along the normal, perpendicular to the mirror. The angle at in each triangle is also equal, because the incident ray meets the mirror surface at , and forces the reflected ray to leave at the same angle on the image side. So triangle is congruent to triangle by the angle-side-angle rule.
Step 5. Read off the result. Congruent triangles have equal corresponding sides, so . Since this holds for any choice of the second ray (any angle ), every ray from appears, after reflection, to diverge from the same point , and that point lies at behind the mirror. The image distance therefore equals the object distance, , which is exactly the property used throughout the worked examples below.
Worked example
A student stands in front of a plane mirror. Find (a) how far the image is behind the mirror, and (b) how far the image is from the student. As a stretch, two parallel mirrors, and , are then set up on opposite walls with a lamp exactly midway between them — describe what is seen and explain why the images eventually vanish.
Step 1. Apply the result just proved: the image forms as far behind the mirror as the object is in front, so the image distance behind the mirror equals the object distance in front, .
Step 2. Find the object-to-image separation by adding the two perpendicular distances, since the object, mirror and image lie on the same straight line: .
Answer: the image is behind the mirror, and from the student.
Competitive stretch
Step 1. Identify the repeated imaging. forms a first image of the lamp; then reflects both the lamp and that first image to form further images, which reflects again, and so on without geometric limit — each mirror keeps re-imaging everything already formed in the other.
Step 2. State the result. This produces an infinite series of images of the lamp, receding into the distance in both mirrors, each pair placed symmetrically about the midline.
Step 3. Explain the fading. No real mirror reflects of the light striking it; a small fraction is absorbed at every reflection. Each successive image in the series has undergone one more reflection than the last, so its brightness keeps dropping, until it becomes too faint to see — even though the geometric series itself never ends.
Answer: an infinite series of images is seen in each mirror, one behind the other, and the series appears to end only because light absorbed at every reflection makes each successive image progressively fainter.
Common traps
- Saying the plane-mirror image can be projected on a screen — it cannot, since it is virtual and no light actually converges behind the mirror.
- Confusing lateral inversion with an upside-down image — a plane mirror image is erect; only left and right are swapped, not top and bottom.
- Assuming the image shrinks or grows as the object moves closer to or farther from the mirror — for a plane mirror the image is always the same size as the object, at every distance.
- Believing two parallel mirrors give only two or three images — properly aligned parallel mirrors give a theoretically infinite, though practically fading, series.
Ask the tutor: Want to see how the multiple-image pattern changes if the two mirrors are tilted instead of perfectly parallel, or work through the congruent-triangle proof with your own numbers? Ask the AI tutor for a guided walk-through.
Why does a mirror flip left and right but not flip me upside down?
A plane mirror does not actually swap left and right in any special way; it reverses whatever direction points straight into the mirror, which is front-to-back, not left-to-right. When you face the mirror, your front becomes your image's front pointing back at you, and because your image is a mirrored version of you along that front-back axis, your right hand ends up on the same side as the image's left hand. Up stays up because nothing about the mirror touches the vertical direction at all.
Why can I never catch a plane mirror image on a screen or piece of paper?
Because the image is virtual: the light rays reflected from the mirror are diverging, spreading apart as if they came from a point behind the mirror, but they never actually pass through that point. A screen can only show an image where reflected or refracted rays truly meet and cross, which is what happens with a real image. Since no light energy is actually present behind a plane mirror, there is nothing there for a screen to capture — the image exists only for an eye or camera positioned to receive the diverging rays and trace them backward.
Do the multiple images in two facing mirrors really go on forever?
Geometrically, yes — each mirror can always form one more reflection of whatever image already exists in the other, so the series has no natural endpoint. In practice you never see anywhere close to infinity, because every real reflection loses a little light to absorption in the glass and the silvering. After enough reflections the image is so faint it falls below what your eye can detect, which is why the visible corridor of images in a barber's shop or dressing room trails off into darkness after a few dozen repetitions rather than continuing forever.