Understanding the Math Behind A: Minimum Length of Plane Mirror

[User1265]

Homework Statement

15. What is the minimum length of a plane mirror in order for you to see a full
view of yourself?
A 1/2 your height B 1/4 your height
C 3/4 your height D your full height

Q Why is answer of A given is the correct one,

Relevant Equations

angle of incidence = angle of reflection

15. What is the minimum length of a plane mirror in order for you to see a full
view of yourself?
A 1/2 your height B 1/4 your height
C 3/4 your height D your full height

Q Why is answer of A given is the correct one,
I understand pictorially how it is, since visually if you were to draw a mirror of the same length as the man, and draw the virtual image produced
you will see the reflected ray of light coming from the feet, emerge halfway up the mirror approximately en route to the real object (the man).
(Image explaining this is attached)

But is there not a more mathematical/geometrical way of explaining how A is so?

 

[kuruman]

Hi @User1265 and welcome to PF!

You can use a geometric description in which there are two similar right triangles. The base of the smaller one is the length of the mirror and its height is the distance from the mirror to the object. The base of the larger one is the height of the image and its height is the distance from the image to the object. Use the relation that you know about similar triangles to derive an expression for the ratio of the bases in terms of the ratio of the heights.

 

[PeroK]

Homework Statement: 15. What is the minimum length of a plane mirror in order for you to see a full
view of yourself?
A 1/2 your height B 1/4 your height
C 3/4 your height D your full height

Q Why is answer of A given is the correct one,
Homework Equations: angle of incidence = angle of reflection

15. What is the minimum length of a plane mirror in order for you to see a full
view of yourself?
A 1/2 your height B 1/4 your height
C 3/4 your height D your full height

Q Why is answer of A given is the correct one,
I understand pictorially how it is, since visually if you were to draw a mirror of the same length as the man, and draw the virtual image produced
you will see the reflected ray of light coming from the feet, emerge halfway up the mirror approximately en route to the real object (the man).
(Image explaining this is attached)

But is there not a more mathematical/geometrical way of explaining how A is so?

Does it matter that your eyes are not actually at the top of your head?

 

[hutchphd]

But is there not a more mathematical/geometrical way of explaining how A is so

Yes there are many (similar triangles, Fermat's Principal, wave interference) but what I carry around in my head for this is essentially a little sketch on crumpled paper. Why do you want a more "mathematical" description?
Just Curious.

 

[PeroK]

Yes there are many (similar triangles, Fermat's Principal, wave interference) but what I carry around in my head for this is essentially a little sketch on crumpled paper. Why do you want a more "mathematical" description?
Just Curious.

Where would we be if Euclid had been satisfied with a piece of crumpled paper?

 

[hutchphd]

Not too crumpled for his work! Perhaps he would have been ready to think about the next really good "mirror" question: why do we appear to be shifted left-right but not upside down?

Of course the joy of physics is that any question, pursued with sufficient vigor and rigor, will open up the entire vista. I do really wonder what manner of explanation was being sought...

 

[PeroK]

Perhaps he would have been ready to think about the next really good "mirror" question: why do we appear to be shifted left-right but not upside down?

By symmetry, if we are shifted left-right, we must be shifted upside down. So, maybe it's neither. Or, have I got all this back to front?

 

[haruspex]

why do we appear to be shifted left-right but not upside down?

It's because of the way we define left and right of people we observe.
The mirror does not really flip left and right; the image of my left is on the left of what I see. But if I interpret what I see as a real other person, the only way to explain what I see is that the person is rotated around a vertical axis relative to me.
In effect, we use a corkscrew rule for this, so LR are defined in terms of UD and FB. If you flip one of UD and FB (the latter being what the mirror does) then it flips our LR too.

Had we no bilateral symmetry, we would not see our reflections as people at all, but some other species.

 

[PeroK]

the only way to explain what I see is that the person is rotated around a vertical axis relative to me.

Why a vertical axis? Why not a horizointal axis?

Is the image really rotated? If someone raises their right hand and is rotated, it's still their right arm that is raised. That wouldn't be a mirror image!

 

[haruspex]

Why a vertical axis? Why not a horizointal axis?

Because that would not match what I see. I see a head at the top, feet at the bottom, an arm at each side, eyes facing me. Because of bilateral symmetry I cannot easily tell that the hand I see on the left is a left hand - it could be either. This allows me to interpret what I see as a person, but only if I interpret the hand I see on the left as a right hand.

Are they really rotated?

No.

 

[PeroK]

Because that would not match what I see. I see a head at the top, feet at the bottom, an arm at each side, eyes facing me. Because of bilateral symmetry I cannot easily tell that the hand I see on the left is a left hand - it could be either. This allows me to interpret what I see as a person, but only if I interpret the hand I see on the left as a right hand.

No.

I would say that, by symmetry, there is no reason that left/right is any different from up/down. In fact, if you lie on your side, then the left/right reflection would be via a horizontal axis.

The explanation I would give is that the reflection changes the direction of front and back. The's the key, because front and back is along the line of the reflection and is the odd dimension. Up/down and left/right relative to the mirror must be symmetrical

Now, left and right of an object are defined relative to front and back. If you take a rectangular block. If you mark one face on the block as "front", then that defines the left and right sides of the block. If you relabel the opposite face as "front", then left and right are redefined.

The image in the mirror is changed from the subject only in that the image is facing in the opposite direction. What is left for the subject is still on the left, but becomes redefined as right for the image - because the front of the subject has been reflected in the image.

 

[haruspex]

The explanation I would give is that the reflection changes the direction of front and back.

left and right ... are defined relative to front and back.

Which is part of what I wrote. The other part is that we define them that way when viewing people because of our bilateral symmetry. It does not apply to objects in general.

 

[PeroK]

Which is part of what I wrote. The other part is that we define them that way when viewing people because of our bilateral symmetry. It does not apply to objects in general.

Okay, but I think it's front and back that is key. As long as we think only of left/right and up/down then there is a paradox there. It's often the case with these puzzles that it's what's left out that is the key. If you give this puzzle to someone and emphasise left/right and up/down, then this distracts them from thinking about the third dimension/direction.

I think it does apply in general. For example, if you take some writing and look at it via a mirror, then it looks right to left. And, if you orient the writing vertically, then it's still the wrong way round in the mirror! There is, therefore, the same thing going on in both these dimensions. I guess that's what you mean by "bilateral symmetry".

The solution must, therefore, be in considering the third dimension; and, in both cases, the writing is actually "back to front". It's not left to right or upside down; it's back to front.

 

[haruspex]

if you take some writing and look at it via a mirror, then it looks right to left.

It looks backwards because you turned it around on a vertical axis in order to see it in the mirror. You could just as easily have flipped it about a horizontal axis to see it; then it would look upside down.

 

[PeroK]

It looks backwards because you turned it around on a vertical axis in order to see it in the mirror. You could just as easily have flipped it about a horizontal axis to see it; then it would look upside down.

That's interesting. I think that might be an alternative way to explain it. It's what you have to do to an object to get it to face the mirror. If someone is standing in front of you, facing you, then to get them to face the mirror, they rotate about a vertical axis. Alternatively, if you rotate them about a horizontal axis, they will be upside down in the mirror.

My way of looking at it was simply that if you imagine a block with a shape cut out, then the shape you see in the mirror image is precisely the shape you see from behind the object. I.e. looking in the direction perpendicular to the plane of the mirror.