Position and Displacement

An object has some position in $3D$ space, and this position can change with motion. One way to identify an object's position is with its $3D$ coordinates $(x,y,z)$. It can also be represented by a vector $\vec{s}$.

The displacement of an object is the signed difference of the position of an object before and after a motion. This is always a straight-line distance, so the displacement of an object that moved from $\vec{s_0}$ to $\vec{s_f}$ is simply:

$$\Delta\vec{s}=\vec{s_f}-\vec{s_0}$$

In terms of coordinates, if an object moves from $(x_0,y_0,z_0)$ to $(x_f,y_f,z_f)$, its displacement is

$$(x_f-x_0,y_f-y_0,z_f-z_0)$$

Position

If we are going to describe motion, we must come up with a way to describe where something is---it's position. Now you may be familiar with describing positions. For example, you might say that your friend is "under the tree", or that the plate is "on the table", and these statements do indeed specify positions to some degree. However, we must devise a way to specify positions more precisely than that. That will be the topic of this lesson.

The first thing to know about positions is that they are strictly relative. There is no way to specify a position without reference another position. For example, if you say your friend is under the tree, you are specifying his position with respect to the tree. If you say a plate is on the table, you are specifying the position with in relation to the table.

Distance

We will represent positions with italic letters, such as $\newcommand{\p}[1]{#1}\p{p}$ or $\p{q}$.

Let's study the relationship between positions. The most basic relationship between two positions is how far apart they are. The technical term for this is the distance between them. Distance is a physical quantity with dimensions of length. If $\p{p}$ and $\p{q}$ are positions, the distance between them is written $|\p{p}-\p{q}|$.

Distances are typically measured with a ruler, as you surely are already aware. A ruler can also be used to add distances. Let's suppose the distance between $\p{p}$ and a point $\p{r}$ is four centimeters and the distance between $\p{r}$ and $\p{q}$ is eight centimeters. Then the sum of the two distances, as you probably already know, is twelve centimeters. To get this distance, you move along the ruler a distance equal to $|\p{p}-\p{r}|$, and then, starting where you left off, you move along the ruler a further distance equal to $|\p{r}-\p{q}|$. Starting at zero centimeters, this process would take you to twelve centimeters. The confusing point is that the sum of the distance from $\p{p}$ to $\p{r}$ and the distance from $\p{r}$ to $\p{q}$ is not just the distance from $\p{p}$ to $\p{q}$. To see why, imagine that $\p{z}$ is in fact just $\p{p}$. Then the distance between $\p{p}$ and $\p{q}$ is zero, while $|\p{p}-\p{r}|+|\p{r}-\p{q}|$ may easily still be greater than zero.

While the distance between $\p{p}$ and $\p{q}$ may be less than the sum of their distances to the point $\p{r}$, it cannot be greater. This fact is known as the triangle inequality. It is expressed as $$|\p{p}-\p{q}|\le |\p{p}-\p{r}|+|\p{r}-\p{q}|.$$

Displacement

Distance is not a very specific measure of the relationship between two points. In particular, there may be two different points, $\p{p}$ and $\p{q}$, that are the same distance away from a third point $\p{r}$, but in different directions. Clearly distance is not sufficient to distinguish the different relationships the $\p{p}$ and $\p{q}$ have with $\p{r}$. To make this distinction, we need a quantity which measure not only the amount of distance between two points, but also the direction. A quantity that represents an amount of distance as well as a direction is called a displacement. The displacement from a point $\p{q}$ to a point $\p{p}$ is written $\p{p}-\p{q}$. Variables representing displacements are written in bold. For example, we could set . A displacement is a physical quantity having dimensions of length.

There are several operations we can do with displacements. One thing we can do is to take the distance associated with a displacement. This distance is called the magnitude of the displacement. The magnitude of a displacement $\v{v}$ is written $|\v{v}|$ or sometimes $v$. If $\v{v}=\p{p}-\p{q}$, then the distance associated with $\v{v}$ is just the distance between $\p{p}$ and $\p{q}$, so $|\v{v}|=|\p{p}-\p{q}|$. Another thing we can do is to take the direction associated with a displacement. The direction of a displacement $\v{v}$ is written $\hat{v}$. To combine a direction and a distance to form a displacement, you can just multiply them: $\v{v}=|\v{v}|\hat{v}$. In fact, you can combine a signed length (which is just a distance that may be positive or negative) with a direction to get a displacement. If a signed length $L$ is negative, then $L \hat{v}=|L|(-\hat{v})$, where $-\hat{v}$ is defined to be the direction pointing in the direction opposite to $\hat{v}$.

Another thing that can be done to displacements is to add them to a point. So if $\p{r}$ is a point and $\v{v}$ is a displacement, then $\p{r} + \v{v}$ represents the position arrived at by starting at $\p{r}$ and moving a distance $|\v{v}|$ in the direction $\hat{v}$.

Displacements can also be added to each other. If $\v{v}$ and $\v{u}$ are both displacements, then the displacement $\v{v}+\v{u}$ represents the displacement gotten by first moving according to $\v{v}$ and then moving according to $\v{u}$ (or doing the two displacements in the opposite order, it doesn't matter). Finally, a displacement can be multiplied by a number. If is $a$ positive real number, and $\v{v}$ is a displacement, the displacement $a\v{v}$ has the direction $\operatorname{sgn}(a) \hat{v}$ (so that the direction is reversed when $a<0$), but its distance is scaled to the value $|a| |\v{v}|$, so that $|a\v{v}|=|a||\v{v}|$.

Cartesian Coordinates

We have now seen how points are related to each other by displacements. We will now use this information to devise a way to specify a position in a way that is easy to understand. To see where the main difficulty in specifying a position lies, remember that a position can only be specified in relation to another position. For example, if you were asked where a grocery store was, you might say it was three blocks west of the hospital, but then the question becomes, "Where is the hospital?" For this reason, it is useful to agree upon a common reference point from which any other position can be found by adding the appropriate displacement. This reference position is called the origin, and we will write it as $o$. Once an origin is chosen, we can describe the point $p$ with the displacement $\v{p}=p-o$. Conversely, if we start with a displacement $\v{r}$, we can obtain a position $r$ by the formula $r=o+\v{r}$. In this way we have reduced the problem of specifying a position to the problem of specifying a displacement.

However, there is still some creativity needed to specify a displacement. The difficulty lies mainly with the direction of the displacement, since we have seen that the magnitude is a length and can be specified with a number and a choice of units. Similar to fixing an origin, we will ease the process of specifying a direction by agreeing to a standard beforehand. However, while we only need to specify one position as an origin, we will need to specify three directions in order to be able to express every direction (in three dimensional space, in general you need as many directions as dimensions of the space). We will write the three directions of the coordinate system as $\hat{e}_1$, $\hat{e}_2$, and $\hat{e}_3$. Then a displacement $\v{r}$ of any direction and length can be written as $r_1 \hat{e}_1 + r_2 \hat{e}_2 + r_3 \hat{e}_3$, where $r_1$, $r_2$, and $r_3$ are signed lengths. For example, if $\hat{e}_1$ was east, $\hat{e}_2$ was north, and $\hat{e}_3$ was up, and the grocery store is three blocks west of the hospital, then the displacement from the hospital to the fifth floor of the grocery store is $-3 \textrm{ blocks } \hat{e}_1 + 4 \textrm{ floors } \hat{e}_3$. To make the notation simpler, the displacement $r_1 \hat{e}_1 + r_2 \hat{e}_2 + r_3 \hat{e}_3$ is more often written $(r_1,r_2,r_3)$ when the coordinate directions are already understood from context.

Putting everything together, if we choose an origin $o$ and three coordinate directions $\hat{e}_1$, $\hat{e}_2$, and $\hat{e}_3$, then we can represent any position $p$ by three signed lengths $p_1$, $p_2$, and $p_3$ through the rule $p=o + p_1 \hat{e}_1 + p_2 \hat{e}_2 + p_3 \hat{e}_3$. The origin $o$ together with the three coordinate directions constitute a cartesian coordinate system. We now see that a cartesian coordinate system gives a systematic way of representing any position by three signed lengths.

Distance and displacement may seem very familiar, but in the context of elementary physics and higher level mathematics, these two terms are completely different.

Displacement is a measure of the length between two objects of a length's path.

Distance is a measure of the total path length of an object.

Not clear enough?

Assume you are moving in a circle of radius "r" meters. If you walk one time around the circle, you are back to where you started. This means that your displacement or ?d is 0 meters. However, you did actually move around the circumference of the circle. This means that the distance you moved around the circle was the circumference of that circle, which was 2&r meters.