It's not only possible to do what you asked, but it's possible to do it without even knowing how far away from the mirror the object is! Let's start with your first example, and I'll explain...
What you need to think about is the rays of light that start at the top and bottom of the object, bounce off the mirror, and come back to your eyes. The points where the light rays hit the mirror are the top and bottom of where the mirror needs to be, since that's where it will look as if the top and bottom of the object are in the mirror.
So let's look at the beam of light that starts at the top of your head. It goes straight towards the mirror, bounces straight off, and comes straight back to your eyes (since they're at the top of your head). So the top edge of the mirror will need to be just as tall as you are. When the light ray from your toes bounces off the mirror, it has to go at an angle, so it'll come back to your eyes. The thing about light bouncing off mirrors is that it always bounces off at the same angle that it hits. So in the drawing, the two angles are the same. Since the two angles are the same, the two triangles that I've drawn will just be identical reflections of each other:
Since the two triangles are identical, the two lengths are the same. So the point that the light hits the mirror at is at exactly half your height. This is the bottom edge of the mirror (where it will look like your toes are in the mirror). And like I said before, the top edge is going to be at exactly your height.
But what about your second example? Does it really matter that your eyes arenít at the top of your head? Well, instead of being at half your
height, the spot on the mirror that the light from your toes hits will have to be at half of your eyes'
height. Again, this will be the bottom edge of the mirror.
But now you have to do the same thing to figure out the top edge. When the ray of light comes from the top of your head, it bounces off the mirror and comes back to your eyes. The spot that it bounces at will be halfway between the height of your eyes and the top of your head. So its total height on the wall will be the height of your eyes plus half of the distance between your eyes and the top of your head. This is the top edge of the mirror:
So in the end, it doesn't really matter how far away you are. All that matters is how tall you are and how tall your eyes are.
(republished on 07/28/06)