Lifting a square tile by one of its corners

[Ocata]

Hi,

Suppose there is a very thin square tile set evenly on the floor such that one corner is at the origin, (0,0) of an xy plane. The tile is 1 x 1 units and thus, the corner of the tile opposite of the origin is at (1,1) on the xy plane.

Suppose the corner of the tile at (1,1) is grasped and lifted by some amount such that the corner of the tile at (1,0) is raised by .25 units and the corner of the tile at (0,1) is raised by .25 units as well.

Does this mean that the corner of the tile, (1,1) has been lifted by an amount of .25 + .25 = .5 units?

 

[phinds]

What do you think? Have you tried drawing it? Tried any calculations?

 

[TonyS]

Try thinking about the problem in terms of vector displacements of the corners of the tile.

 

[Ocata]

What do you think? Have you tried drawing it? Tried any calculations?

I think it would work. I have tried drawing it, but I'm not 100% confident that I am accurately illustrating the scenario described.

Is there a non calculus calculation, proof, derivation or someway to confirm this if true?

Thank you.

 

[phinds]

I think it would work. I have tried drawing it, but I'm not 100% confident that I am accurately illustrating the scenario described.

Is there a non calculus calculation, proof, derivation or someway to confirm this if true?

Thank you.

This does not require any calculus, just simple algebra, and no, your assumption of simple addition does not work. You need to think through the geometry, then apply algebra

 

[TonyS]

I think it would work. I have tried drawing it, but I'm not 100% confident that I am accurately illustrating the scenario described.

Is there a non calculus calculation, proof, derivation or someway to confirm this if true?

Thank you.

Visualise vector displacements along the edges of the tile and think about what happens to those vectors when the corner of the tile is lifted.
Then think about how you could use those vector displacements to move from (0,0) to the diagonally opposite corner.

 

[Ocata]

May I adjust a part of the original scenario.

In the original post, I mentioned that the tile is 1 x 1 units. But if that were the case, then when I lift the tile by the opposite corner, the coordinate of that corner would no longer be (1,1,z), but some coordinate (less that 1, less than 1, z). I would prefer to study a situation such that upon being lifted at the opposite corner, the new coordinate of the tile is (1,1, z).

So perhaps I should start with a tile larger than 1 x 1.

For instance. Suppose I have a tile 5 x 5 units. Then the corner at (5,5,0) is lifted by some amount such that the tile has coordinates (1,0, .25), (0,1,.25), and (1,1, z). Therefore, what I'm trying to figure out is, what is z?

 

[Ocata]

Visualise vector displacements along the edges of the tile and think about what happens to those vectors when the corner of the tile is lifted.
Then think about how you could use those vector displacements to move from (0,0) to the diagonally opposite corner.

Okay, I will give it a try.

 

[phinds]

May I adjust a part of the original scenario.

In the original post, I mentioned that the tile is 1 x 1 units. But if that were the case, then when I lift the tile by the opposite corner, the coordinate of that corner would no longer be (1,1,z), but some coordinate (less that 1, less than 1, z). I would prefer to study a situation such that upon being lifted at the opposite corner, the new coordinate of the tile is (1,1, z).

So perhaps I should start with a tile larger than 1 x 1.

For instance. Suppose I have a tile 5 x 5 units. Then the corner at (5,5,0) is lifted by some amount such that the tile has coordinates (1,0, .25), (0,1,.25), and (1,1, z). Therefore, what I'm trying to figure out is, what is z?

[bolding mine] I think you mean not that "the tile has ... " but some point ON the tile has ..." and you have not simplified the problem at all. You're making this much more difficult than it needs to be.

 

[TonyS]

May I adjust a part of the original scenario.

In the original post, I mentioned that the tile is 1 x 1 units. But if that were the case, then when I lift the tile by the opposite corner, the coordinate of that corner would no longer be (1,1,z), but some coordinate (less that 1, less than 1, z). I would prefer to study a situation such that upon being lifted at the opposite corner, the new coordinate of the tile is (1,1, z).

So perhaps I should start with a tile larger than 1 x 1.

For instance. Suppose I have a tile 5 x 5 units. Then the corner at (5,5,0) is lifted by some amount such that the tile has coordinates (1,0, .25), (0,1,.25), and (1,1, z). Therefore, what I'm trying to figure out is, what is z?

If you start out with a tile cornered at the origin in the xy plane then lift the corner that was lying at (5,5,0), the corners that started out at (5,0,0) and (0,5,0) will certainly not have the x and y coordinates that you have assigned in your modification of the problem.
Cut out a square of cardboard and try it for yourself !

 

[jbriggs444]

Is there a non calculus calculation, proof, derivation or someway to confirm this if true?

Rather than concentrating on coordinates, you could concentrate on the geometry. In the original version, if the two corners are lifted by 0.25 units each then a line running between them will be horizontal. It will be 0.25 units above the x y plane throughout its length.

A diagonal line running from the corner at the origin to the corner that has been lifted will intersect this horizontal line and will, therefore, be 0.25 units above the x y plane at that point. By symmetry, the two halves of the diagonal line must be of the same length. Because they are part of the same line, they must have the same slope. So the corner that has been lifted must be how far above the plane?