Let's learn about Position, Displacement, and Distance! These are essential concepts that need to be properly understood for further physics studies!

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If we want tell someone where the cookie jar is, it’s not very helpful to say “Oh, it’s four meters to the left” if they don’t know where we are and what we mean by “left”. So the first thing we need to do is agree on a reference frame. To do that, we need to choose a reference point as well as coordinate axes. We call that reference point the origin, since it’ll be the origin of our coordinate axes. Alright, let’s go ahead and choose one now. In a 1-dimensional space, it’s simple to define coordinate axes, because we need only one. And look, now we can talk about left and right. We can also do this in 2D space. Let’s add an extra axis, like so, and now we’ve got up and down as well. And in 3D, we would need a third axis, like so, and now we’ve got front and back.

Now that we’ve got a reference frame, it does make sense to say “Oh, it’s four meters to the left”. And that brings us to the most obvious thing we can do with a reference frame: describing the positions of things!

So, let’s take a look at an example. Here is a 1-dimensional reference frame. If I tell you that a particle is four meters to the left of the origin, you now know exactly where it is: right there.

I want you to notice something though. I had to give you two pieces of information in order to specify the position of the particle. I had to say that it’s four meters away from the origin, and that it’s to the left of the origin. In other words, I had to tell you its distance as well as its direction relative to the origin. Those two quantities, distance and direction, together specify the position.

Here’s a good time to make a brief side trip into mathematics.

A mathematical quantity that’s described by a magnitude and a direction is called a vector. Vectors are kind of like arrows: they have a certain size or length, and they point in a certain direction. If this concept isn’t familiar to you, then we suggest that you watch our lesson on “Introduction to Vectors”, for more context to this lesson.

So what’s this got to do with the position of a particle? Well, when specifying the position of something, we need to mention how far it is, AKA the distance, which is a magnitude, and the direction it’s pointed in. So since position consists of both the properties required to describe a vector, we can say that position is a vector!

Now you’re probably wondering: couldn’t we choose to represent left with negative numbers and right with positive numbers? Of course we could. Then the particle we mentioned before would be at position negative 4 meters.

In two dimensions, though, direction is not so easily specified. For example, here’s a particle located 1 meter south and 2 meters east of the origin. We can’t represent the direction by a single sign in this case. But what we could do is declare that north and east are positive, and that south and west are negative. Then we could represent the position as a list of numbers: (2,-1). In fact, this list of numbers is just the coordinates of a point on a plot.

Great!

So now we know how to talk about the position of things relative to the origin. Now, what if an object moves?

That brings us to the concept of displacement. Let’s take a look at an example here. A particle, initially located at negative 3 meters, moves to a new position at positive 4 meters. We can ask, what’s the final position of the particle relative to its initial position? That’s easy: it’s 7 meters to the right of the position it started at. It’s easy to see this, but we can calculate it too. Simply take the final position and subtract it by the initial position to give us 7 in this example. So, that gives us the displacement of 7 meters to the right.

Now, one thing we need to remember about displacement is that it doesn’t matter how the particle moves from one position to another. We’re only concerned with where it started, where it ended up, and the relative positions of the two.

Here’s another way to think about displacement. It’s simply the final position of the particle when we move the origin to its initial position. So if we move the origin 3 meters to the left, we find that the final position of the particle is 7 meters to the right of the new origin.

Good!

Now, let’s go ahead and look at a 2D example.

Here’s a particle that’s 1 meter north and 3 meters west of the origin. Its coordinates are (-3,1). It moves to a new position: (5,-2). What’s its final position relative to its initial position? It seems complicated, but let’s break it down. In the east-west direction, the particle starts at negative 3 meters and ends at positive 5 meters. Again, like we mentioned, we can find the displacement of this by taking the final position and subtracting it by the initial position. 5 minus (negative 3) equals 8: that’s a displacement of 8 meters east. Good.

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