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I have defined velocity in forward direction to be positive and negative in the backward direction. So what is the velocity if the body is moving right or left?

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    $\begingroup$ You've obviously heard of vectors, so it's not clear to me what you're actually asking here. You could use imaginary numbers for left - right motion, if the motion is restricted to a plane. $\endgroup$ – PM 2Ring May 1 at 14:06
  • $\begingroup$ Is the OP asking about a "natural" definition of left and right? Since this universe has chirality, there's an answer. $\endgroup$ – Fattie May 2 at 0:08
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    $\begingroup$ @PM2Ring Complex numbers have nothing to do with it. Bringing up $\mathbb{C}$ when someone is really looking for $\mathbb{R}^2$ will only cause confusion. $\endgroup$ – user76284 May 2 at 0:14
  • $\begingroup$ @user76284 I wasn't actually suggesting that it's a good idea, that's why I used italics. And I'm still waiting for the OP to clarify the question... $\endgroup$ – PM 2Ring May 2 at 0:16
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In general velocity is a vector. In 1D motion we can get away with saying the velocity is positive or negative by associating each sign with a direction. However, once you move in more dimensions you can't say the velocity is positive or negative. You just have the vector (in Cartesian coordinates)

$$\mathbf v=v_x\,\hat x+v_y\,\hat y+v_z\,\hat z$$

Of course, each component $v_i$ can be positive or negative, just like you do in 1D motion. But the vector itself can't be given any sign, just a direction.

So, in your case let's make forward and backwards movement in the $y$ direction be positive and negative respectively, and let's make right and left movement in the $x$ direction positive and negative respectively. Then forward means $v_y>0$, backwards means $v_y<0$, right means $v_x>0$, and left means $v_x<0$. With the exact values of these components you can determine the direction of $\mathbf v$.

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The 3D Cartesian Axes are usually defined as Z 'upwards' as positive Z direction, Y 'right' as positive Y direction and X 'out-of-the-page (towards you)' as the positive X direction.

2D obviously the Y and Z axes swap, there is no Z axis, and the X axis 'becomes the 3D Y axis' in effect.

Each axis is independent from each other so have their own positive and negative direction. If the object moves along multiple axis, you need to define positive and negative along each axes (that the object moves along). E.g.Moving in 2D along X and Y: $$ \underline{v} = \begin{pmatrix} 1\\ -2 \end{pmatrix} ms^{-1} $$ in matrix notation. So the object moves 1 metre in the positive y direction, and 2 metres in the negative x direction every second.

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    $\begingroup$ In my experience the usual convention is +x to the right +y to the top of the page and +z out of the page. The nice thing about this approach is that you don’t have to swap axes for 2D vs 3D. Instead x and y stay the same and you just add or remove z. $\endgroup$ – Dale May 1 at 14:27
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    $\begingroup$ @Dale In my experience, people like $z$ for the symmetry-breaking direction. Beam physicists like $z$ to point "downstream" and $x$ to point "beam left." People doing work with magnets like uniform fields to point along the $z$ direction, and quantize atomic spins along the $z$ axis. People who are doing Newtonian physics in Cartesian coordinates like $z$ to be vertical, along the gravity direction. It's so easy to relabel that folks usually don't worry too hard about stumbling into a different convention. $\endgroup$ – rob May 1 at 17:40
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    $\begingroup$ @rob yes, I also don’t worry about different conventions. I just wouldn’t describe the one used here as “usually defined”. My idea of “usually defined” is different and as you mention many different specialties have adopted their own distinct usual conventions. $\endgroup$ – Dale May 1 at 18:02
  • $\begingroup$ All the relabelling is, is a rotation of the axis, no matter which axis you label as which. I only gave usually as its the most common choice that I have seen in my experience. But as you say, you are free to choose them as you wish, as long as you are consistent! Indeed using the right hand rule would suggest that the X axis be into the page, for clockwise rotations. (clockwise being the convention) but that is much hardly to pictorially represent! $\endgroup$ – SamuraiMelon May 1 at 21:37
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    $\begingroup$ @rob When working with lens for glasses, y positive is up, but x positive is "towards the nose". So, for the right eye, x is positive to the right, but for the left eye x is negative to the right. Also, since you are viewing it from the doctor side, right is to the left, and left is to the right (Just to point out a strange labeling convention) $\endgroup$ – vals May 2 at 8:41
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I think it would help to read up on the definition of a vector.

Velocity, Acceleration, Force, and many other quantities in Physics are described by vectors.

From a geometrical point of view most texts begin with the idea of a directed line segment. Just as we have the idea of a line segment from points P to Q, where only the length has meaning, we define a directed line segment from P to Q, which has magnitude and direction. The directed line segment from Q to P points in the opposite direction but has the same length. For these quantities the starting and ending points matter.

The mathematical definition of a vector is an equivalence class of all directed line segments with the same length and direction. This is a vector.

Vectors are usually referenced relative to a set of coordinate axes. In two dimensions you can draw them on graph paper with the usual x and y axes defined. In three dim you need x, y and z. You can think of them in pairs as (forward and backward) along the x axis, (left and right) along the y axis, and (up and down) along the z axis.

A velocity vector (any vector for that matter) will have three components and can be written as $(V_x, V_y, V_z)$, just like a point although there are distinct differences so sometimes other notations are used, e.g. $<V_x, V_y, V_z>$.

Now you can make sense of the sign change. The original vector defines a direction, and the opposite direction is just $<-V_x, -V_y, -V_z>$. If an object changes only its forward component, while moving in the other directions this cane be expressed as $<-V_x, V_y, V_z>$, changing the up direction of motion to down is just $<V_x, V_y, -V_z>$, and changing left to right (or vice verse) is $<V_x, -V_y, V_z>$. If an object is moving in a straight line along only one of those directions then its vector is just, $<V_x, 0, 0>$ for forward-backward, and $<0, V_y, 0>$ for left-right, etc. In these cases the motion is simpler and we don't really use vectors.

I hope this helps a little.

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All of the answers about vectors and matrices seem technically correct, but I am not sure that that is what is being asked here. If I understand the question correctly, the short answer is "it is entirely arbitrary". However, there is a convention.

If you only have one direction then it is conventionally labeled "the x axis". It is conventionally drawn horizontally, with positive to the right, like this:

-ve <----------> +ve (x)

If you have a second direction then it is conventionally labeled "the y axis". It is conventionally drawn vertically, with positive pointing up the page, like this:

+ve (y)
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 v
-ve

To answer your question: if the x direction is forward and backward with forward positive, then by convention the y direction is left and right with left positive.

To further develop your question: by further convention the z direction would be up and down with up (poiting toward you, out of the screen) positive.

This is how it is typically shown in maths textbooks. Other platforms might use different depictions, though the relative signs of x, y and z will always be the same. For further reading, see "right hand rule": https://en.wikipedia.org/wiki/Right-hand_rule#Coordinates.

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