# How do I know if a motion is 1 dimensional or 2 dimensional?

If an object is moving in a straight line with an angle with x axis (it may be vertical or horizontal) , is it 1 dimensional or 2 dimensional?

The question was asked by my teacher and he himself gave the answer next day that it is a rectilinear motion . His reason was that the motion is still in a straight line (i.e., the object can only move forward or backward along that direction).

But I am still not able to understand it. How can it be a 1D motion when on moving , it's two coordinates are changing. (x and y if vertical angle) or (x and z if horizontal angle).

• A straight line is always one-dimensional, because there is only one direction you can go: along the line. You therefore need only one number du specify where you are on the line (after choosing a point on the line to be $0$), hence one-dimensional. It does not matter how many dimensions the space in which the line is embedded has. Things get more complicated when you are dealing with curved lines, those usually need higher-dimensional spaces to be properly embedded. But questions about this are better suited to math stack exchange. Commented Jul 18 at 21:23
• You can change the coordinate system so that the x-axis is along the line and the y-axis is perpendicular to it. That does not change the motion, it only changes how you describe it. Commented Jul 18 at 21:30
• You can choose a coordinate system in which one or two or three coordinates change as a particle moves along the line, but if the particle cannot leave the line, then some people would say that it has only one degree of freedom. That's because of what Paulina said. You can fully specify the particle's position with just one number—its distance from some reference point on the line. Commented Jul 19 at 2:47

Let's say you have the following motion (the red arrow)

It looks 2D on this plot, since its motion changes both the $$x$$- and $$y$$-coordinates. However, if you redefine the axes like so:

Then the motion is 1D. Only one coordinate changes.

Granted, the new coordinates are not the same as the old coordinates. If you measure the movement using your old coordinates, then it looks 2D. It's just that with an appropriate choice of coordinates, you can make the motion 1D.

• And to highlight: changing co-ordinates does not change the motion — it only changes how you describe the motion.  So if the motion is one-dimensional with respect to one co-ordinate basis, it's one-dimensional with respect to any basis.  Picking an appropriate basis simply makes that more obvious. Commented Jul 19 at 11:05

Both you and your teacher can be correct depending on what you mean by "dimension."

In everyday experience, we generally consider dimensionality to be the number of independent parameters needed to describe the position of an object within the universe. This is why people say that we live in a 3D or 4D universe (depending on whether you want to put time and space on equal footing).

In certain fields of math and physics (especially linear algebra) however, dimensionality refers to the number of parameters required to describe a certain object or motion. For example, the motion of a mass (released from rest) on a spring is one dimensional because it is entirely described by $$r(t)=A\cos(\omega t).$$ But this doesn't change anything about the spatial dimensions of the universe - the spring is still in 3D, it's just that its motion is simple enough to analyze with just one component. We could say that "the one dimensional motion of the spring is embedded within a three dimensional universe." Similarly, if you released the mass and gave it a little kick in an arbitrary direction, its motion would be $$\mathbf{r}(t)=(A\cos(\omega t+\varphi_1), B\cos(\omega t+\phi_2))$$ which we would call two dimensional. But I hope it's clear that doing this doesn't change that we live in a 3D spatial universe.

One dimensional motion is any kind of motion that happens on a line. There are many ways to define this. For example, you could say that the position vector of the particle is always $$\vec{r}(t)=r(t)\hat{n}$$ for some unit vector $$\hat{n}$$. In other words, the motion is always along the ray defined by this vector. This is in contrast to two dimensional motion that requires at least two vectors to define the set of points that the particle covers. In the particular case you mention, you can also rotate the axis such that it aligns with the direction of motion of the particle, making it trivially one dimensional.

Sometimes, motion may appear two dimensional or three dimensional when it really isn't. For example, think of a particle moving on a circular path. Is the motion one dimensional, or two dimensional? From a cartesian reference frame, the motion seems two dimensional: after all, you are defining it in terms of the $$\hat{x}$$ and $$\hat{y}$$ vectors. But that answer is wrong. In polar coordinates, the path of the particle is simply $$\vec{r}=a\hat{r}$$ for a circle of radius $$a$$, where $$\hat{r}=\cos(\theta)\hat{x}+\sin(\theta)\hat{y}$$. As you can see, in polar coordinates the motion is written in the same fashion as the definition in the above paragraph for $$\hat{n}=\hat{r}$$ and $$a=r(t)$$.

What is ultimately important is whether the particle is covering a surface or not. A circumference can be "straightened out" into a straight line. A circle cannot because it's a qualitatively different object.

• Since the last paragraph is focusing on a surface, should it be referring to a disc as opposed to a circle? Commented Jul 19 at 6:44
• @Flater At least in my language, a "circle" can refer also to the internal surface within a circumference, while the circumference simply refers to the outline. Commented Jul 19 at 8:33
• I would argue that one dimensional motion is motion where the trajectory can be defined by means of a single parameter - which is not possible for a surface or a volume or beyond. Commented Jul 19 at 11:48
• The circle example is actually kinda tricky because there's a difference between a particle that is constrained to move in a circle and one that can move in more dimensions and just happens to move in a circle. After all, every path is one-dimensional; the question is really about the dimension of configuration space. Commented Jul 19 at 13:18
• @Agog0615 I'm not sure how talking about polar coordinates is supposed to help you here. When discussing circular motion, it's important for you to understand that it always involves acceleration even if speed is constant. That's because acceleration is a change in velocity and direction is a component of velocity (but not of speed.) If your teacher is talking about a something moving at a constant velocity, you can be sure it is moving in a straight line and not a circular path. Commented Jul 19 at 19:39

the main thing take away: physics does not care a about your coordinates.

So if the thing is moving on:

$$\vec r(t) = t\cos{\theta}\hat e_1 +t\sin{\theta}\hat e_2$$

thats linear motion in a plane. Now I can also describe it as linear motion in a hyper$$^3$$-cube:

$$\vec r(t) = t\cos{\theta}\hat e_1+ t\sin{\theta}\hat e_2 + c_3\hat e_3 + c_4\hat e_4 + c_5\hat e_5 + c_6\hat e_6$$

which doesn't really make it six dimensional.

An optimal coordinate choice is:

$$\hat x = \cos\theta \hat e_1 - \sin\theta \hat e_2$$ $$\hat y = \sin\theta \hat e_1 + \cos\theta \hat e_2$$

and the motion is now:

$$\vec r(t) = t\hat x$$

Definitely one dimensional.

Now if I put it in a 2-D rotating frame, so there are 2D Coriolis and Centrifugal forces, which cancel...then it might be 2D.