Felice R. | Need Help in Math and Science - Engineer willing to helpNeed Help in Math and Science - Engineer...

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this would be true.

By definition a plane is a flat surface that has no thickness and extends forever, if we take a cube we know that the top and the bottom of the cube are parallel to each other, so if in 3D space we have a cube that is 2x2x2.

The bottom of the cube would we lying in a plane that is defined by z = 0, and its top is lying in the plane where z = 2. We know that the top and bottom of the cube are parallel to each other, so the planes that contain the top and the bottom are also parallel to each other. They will not intersect

Michelle raised a good point. You can write the equations of two planes in the Hessian normal form, i.e., for example

2x+3y-1=0

2x+3y-2=0

The two equations defined two planes in R3, and you can easily verify that they have no point in common. Hence, in Euclidean geometry, they must be parallel.

Well, true. You could show (in Euclidean geometry) that two planes that are perpendicular to the same line are parallel planes.

I stressed the fact that this is true only in Euclidean geometry. In fact, recall that the existence of parallel lines in Euclidean geometry is one of the postulates and
nota theorem, i.e., something that we decide is true as a rule for the game. There are other geometries (hyperbolic, elliptic) where this is not the case.

Since nobody has ever followed two infinite parallel lines to check whether or not they really never cross, it is legitimate to assume that there could exist other valid geometries besides the Euclidean one.

This is more than just playing a game of "what if". So much so that non Euclidean geometries have application in one of the most genial theory in the history of humanity: the general relativity theory by Albert Einstein according to which space is "curved". But that is a story for another time. Geometry is a fascinating and beautiful field, not to mention useful as well.

James A. | Middle School Math and Science TutorMiddle School Math and Science Tutor

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True, so long as your subject matter is including the possibility of a 3 dimensional workspace. For instance, you can have 2 dimensional planes that are parallel to each other in a 3 dimensional graph. Hope this helps.

## Comments

Michelle raised a good point. You can write the equations of two planes in the Hessian normal form, i.e., for example

2x+3y-1=0

2x+3y-2=0

The two equations defined two planes in R3, and you can easily verify that they have no point in common. Hence, in Euclidean geometry, they must be parallel.