# Why are dimensions always at right angles?

I asked my teacher , i didnt get any satisfactory answers , can u tell me why dimensions are always at right angles .

#### Giant

Not true at all. Dimensions aren't always at right angles. They can be whatever they want.
Not true at all. Dimensions aren't always at right angles. They can be whatever they want.
We *choose* 90 degrees because a lot of problems are avoided by doing so. For example, assume that 'a' is the angle which the axes make, then the distance formula changed to d = square_root((x2 - x1)^2 + (y2 - y1)^2 + 2*(x2 - x1)*(y2 - y1) * cos(a)); Now by choosing a=90 degrees we eliminate the cos term and get a beautiful formula, instead of an overcomplicated one. Similarly a LOT of trigonometric terms are avoided, sin(90) becomes 1 as well. If complications don't arise and instead stuff gets simplified, mathematicians do indeed take axes which are inclined at an angle.

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#### HallsofIvy

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I will also point out that "dimension" is the wrong word here. "Dimension" is a number; what you are calling "dimensions" are the coordinate axes. As others say here, they are not necessarily at right angles. They are often chosen to be at right angles because that simplifies the calculations.

#### Hornbein

I asked my teacher , i didnt get any satisfactory answers , can u tell me why dimensions are always at right angles .

There all all sorts of spaces. In some of those spaces, dimensions are not at right angles. But the great majority of people are not interested in these, because in our Universe the physical dimensions ARE at right angles. This is true of any space with the Euclidian metric of a^2 + b^2 + c^2 = d^2, where d is the distance between two points and a, b, and c are at right angles to one another.

Some responders are trying to tell you that the right angles are just a convention and convenience. It is possible to use coordinate axes that are not at right angles, but as far as I know no one actually does this in the real world. The real world really does have this right angle character.

#### FactChecker

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The real world really does have this right angle character.
I'm afraid I have to disagree. The real world does not measure things in coordinate systems. People do. The advantage of measuring coordinates in right angles is so great that there are few exceptions. One example is magnetic North/South versus polar North/South. Magnetic North does not line up with Polar North. So if you go in the magnetic North direction, you are (usually) also moving in the East/West direction. Because of this, Latitude / Longitude measurements are not based on magnetic North even though that is the easiest thing to determine in the "real world". But humans made that decision, not the "real world".

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#### Hornbein

I'm afraid I have to disagree. The real world does not measure things in coordinate systems. People do. The advantage of measuring coordinates in right angles is so great that there are few exceptions. One example is magnetic North/South versus polar North/South. Magnetic North does not line up with Polar North. So if you go in the magnetic North direction, you are (usually) also moving in the East/West direction. Because of this, Latitude / Longitude measurements are not based on magnetic North even though that is the easiest thing to determine in the "real world". But humans made that decision, not the "real world".

#### micromass

Well, dimensions are just numbers. So dimensions being at right angles makes no sense really.

#### Hornbein

Well, dimensions are just numbers. So dimensions being at right angles makes no sense really.

A mathematician can say that dimensions are just numbers related in arbitrary ways. That's valid, but physicists deal with real objects and measurements inside of a real universe that is overwhelmingly preferential to 3 dimensions of space and one of time, all at right angles to one another. In the context of physicsforums such may be assumed.

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#### pwsnafu

A mathematician can say that dimensions are just numbers related in arbitrary ways. That's valid, but physicists deal with real objects and measurements inside of a real universe that is overwhelmingly preferential to 3 dimensions of space and one of time, all at right angles to one another. In the context of physicsforums such may be assumed.
This is the mathematics section of the website, hence people are answering in the context of mathematics. By that logic do you assume that the Earth Sciences forum or the Computing forum are also physics discussions just because the website is named Physics Forums?

#### pbuk

When dealing with "real objects", in what sense does time form an angle of 90 degrees with any of the dimensions of space?

#### Giant

When dealing with "real objects", in what sense does time form an angle of 90 degrees with any of the dimensions of space?
He's talking about relativity. It assumes a 4 dimensional space where time is another dimension. Time dilation tell us that time doesn't flow at equal rate depending on your velocity and gravitational potential. So time rate has to be measures. This is simplified by choosing a 4 dimensional space. Time does indeed flow at different rates. Experiments have been done and it's verified.

#### micromass

He's talking about relativity. It assumes a 4 dimensional space where time is another dimension. Time dilation tell us that time doesn't flow at equal rate depending on your velocity and gravitational potential. So time rate has to be measures. This is simplified by choosing a 4 dimensional space. Time does indeed flow at different rates. Experiments have been done and it's verified.
OK, how do you measure 90° angles in GR?

#### Hornbein

This is the mathematics section of the website, hence people are answering in the context of mathematics. By that logic do you assume that the Earth Sciences forum or the Computing forum are also physics discussions just because the website is named Physics Forums?
You've got a point there.

The question makes sense from the point of physics but not from the mathematics side. So I'd charitably assume that he was asking from the physics point of view.

#### Hornbein

When dealing with "real objects", in what sense does time form an angle of 90 degrees with any of the dimensions of space?
It isn't 90 degrees, but it is orthogonal. The Minkowski metric is x^2+y^2+z^2+it^2, which is pretty similar to the Euclidean metric. So one may loosely think of the angle as begin 90 degrees.

#### micromass

It isn't 90 degrees, but it is orthogonal. The Minkowski metric is x^2+y^2+z^2+it^2, which is pretty similar to the Euclidean metric. So one may loosely think of the angle as begin 90 degrees.

Are you sure the imaginary number $i$ should be in there?

And what about vectors for which this metric is zero. Should you interpret it as being orthogonal on itself?

#### Hornbein

Are you sure the imaginary number $i$ should be in there?

And what about vectors for which this metric is zero. Should you interpret it as being orthogonal on itself?
Yep. Albert himself sometimes used that notation. It can also be written as x^2+y^2+z^2-t^2. So on second thought, it should have been x^2+y^2+z^2+(it)^2 with standard operator precedence.

The lines for which the metric is zero are null lines. These are the lines traveled by light in a vacuum. As far as light is concerned, it takes zero proper time to travel anywhere in a vacuum. So it is a pseudometric.

If you are interested, you are invited to look up Minkowski spacetime. I'm sure that there are many others who can explain this better than can I. If that isn't enough, there is the Wick rotation.

#### micromass

Yep. Albert himself sometimes used that notation. It can also be written as x^2+y^2+z^2-t^2. So on second thought, it should have been x^2+y^2+z^2+(it)^2 with standard operator precedence.

The lines for which the metric is zero are null lines. These are the lines traveled by light in a vacuum. As far as light is concerned, it takes zero proper time to travel anywhere in a vacuum. So it is a pseudometric.

If you are interested, look up Minkowski spacetime. I'm sure that there are many others who can explain this better than can I.
I know Minkowski spacetime. I just disagree you can meaningfully talk about angles and orthogonality there. And when you move to GR it becomes even more problematic.

#### pbuk

It isn't 90 degrees, but it is orthogonal.
Only if you choose a particular definition for the inner product. Coordinate systems do not need to be orthogonal, but they do need to be linearly independent.

Which, getting back to the OP, is the point - it is meaningless to talk about dimensions being at right angles, however the coordinate systems we choose to describe position in any geometry are usually more convenient if their bases are orthogonal which, in the case of spatial coordinates, does imply they are at right angles.

#### Cruz Martinez

A mathematician can say that dimensions are just numbers related in arbitrary ways. That's valid, but physicists deal with real objects and measurements inside of a real universe that is overwhelmingly preferential to 3 dimensions of space and one of time, all at right angles to one another. In the context of physicsforums such may be assumed.
How does this make any sense if we are to accept mathematics as the language of physics? The dimension of spacetime (Minkowski a special case) is defined as the number of parameters needed to specify an event biunivocally in a sufficiently small neighborhood. So the dimension is 4, nothing more, how is it meaningfull to mention angles here? Also, a mathematician would not say dimensions are ''numbers related in abitrary ways''.

#### Giant

OK, how do you measure 90° angles in GR?
Ok I thought orthogonal meant 90 degrees. Sorry about that. But orthogonality test would be dot product? I'm not well read about relativity so I wont make further arguments.

#### weirdoguy

Albert himself sometimes used that notation.
But physicists don't use it for several decades now, it's problematic - it's been discussed in Relativity forum multiple times.

#### Mark Harder

Gold Member
Ok I thought orthogonal meant 90 degrees. Sorry about that. But orthogonality test would be dot product? I'm not well read about relativity so I wont make further arguments.
Yes, two vectors are orthogonal if their dot product is zero, by definition. But orthogonal and 90° need not be the same. For one thing, the angle refers to a geometric situation, where the unit vectors are directions in space. Angles have no meaning in other abstract notions of vector spaces. Secondly, one could define different types of "dot" products, for which orthogonal directions are not 90°. A physical and geometric example of this would be a crystal. As you might guess from looking at different minerals that have different shapes, quartz and salt (halite) for example, not all crystals have natural coordinate systems in which the axes meet at 90°. The periodic structures within the crystal, known as unit cells, consist of just enough atoms or molecules so that when the unit cell is translated in 3 directions by the lengths of the unit cell sides in those directions, the entire crystal is generated; and these 3 directions need not be at 90° to each other. It's analogous to periodic functions, like trig functions, that repeat themselves over every period along the axis of the independent variable.

#### Giant

Yes, two vectors are orthogonal if their dot product is zero, by definition. But orthogonal and 90° need not be the same. For one thing, the angle refers to a geometric situation, where the unit vectors are directions in space. Angles have no meaning in other abstract notions of vector spaces. Secondly, one could define different types of "dot" products, for which orthogonal directions are not 90°. A physical and geometric example of this would be a crystal. As you might guess from looking at different minerals that have different shapes, quartz and salt (halite) for example, not all crystals have natural coordinate systems in which the axes meet at 90°. The periodic structures within the crystal, known as unit cells, consist of just enough atoms or molecules so that when the unit cell is translated in 3 directions by the lengths of the unit cell sides in those directions, the entire crystal is generated; and these 3 directions need not be at 90° to each other. It's analogous to periodic functions, like trig functions, that repeat themselves over every period along the axis of the independent variable.
Wow. crystal example makes it a lot clear. Thanks thanks!

#### suremarc

I suppose that spacetime is "orthogonal" in the sense that the metric tensor has a smoothly varying orthogonal eigenbasis. But that's true for any pseudo-Riemannian manifold, so it's more a statement about a class of topological structures than the physical world.

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