Since we will be describing motion in multiple directions — and since such motions often require two or three independent spatial coordinates — our description of position is a different kind of mathematical quantity from a number, even one with units of distance.

To specify a position we have to give three numbers. For example, if you were in New York, you might say to your lunch date, "Let's meet at the corner of 5th Avenue and 42nd street." (2 coordinates) Of course you could say, "Let's meet just in front of the elevators on the third floor of the Times Building at the corner of 5th Avenue and 42nd Street." (3 coordinates) (And really, since you are not just specifying a location but an event, you would have to tell your date when to meet you there: 4 coordinates.) Let's look at the mathematical model we are using to describe location in a bit more detail.

Our mathematical model is now not just a mapping of a point in space into a single number, but into a set of numbers. If we've chosen an origin and axes, we can use our first coordinate as specifying the location along the x-axis, the second along the y, just like when we drew a graph for the eye. When we are talking about position-position graphs, we typically specify the location as the displacement from the origin (considered fixed) and write the pair of numbers this way: (x, y), where x and y are distances (with units).

As long as we are talking about spatial coordinates, we tend to choose the symbols x, y, and for the three coordinates involved).

Position Vectors

But with our x-y plot — a graph for the eye — the 2D plot is really supposed to stand for an image of physical space — a map. It's sometimes useful to think of a point in space as a single displacement from the origin, and represent this as a new kind of mathematical object — a position vector: an arrow going directly from the origin to the position specified by the coordinates (x, y). We call this arrow a vector.

A vector is more than just a pair of numbers; it has a physical meaning: the displacement from the origin in a particular direction for a particular distance. Once we have that idea, we can represent that same vector in lots of different ways. We could rotate the coordinate system around the origin. We could choose a different scale to measure distances with. But the position vector — the direction and distance of the point in space we are interested in from the origin — would remain the same. (Unless of course you move the origin of your coordinate system! This is why we have to fix our origin first.)

A good way to think about physically what a vector means is a real arrow. The tailfeathers of the arrow are tied to our origin, the length of the arrow represents the distance from the origin, and the tip represents the actual position of the point in space we are considering. In other words, the vector really describes the displacement needed to get from the origin to a particular position in space.  

Although it can be confusing, we typically do NOT distinguish between the point at the tip of the vector — the position — and the vector that points to it from the origin. We might say something like "my position is given by this vector." 

We identify in symbology that a quantity has a direction as well as a position by putting a little arrow over it like this: $\overrightarrow{r}$. (Sometimes in textbooks vectors are shown by writing the symbol in boldface.)

To make the connection between the geometrical picture and the algebra we will need to do calculations, we introduce a notation that allows us to include direction in our algebraic representation of equations. This is going to turn out to be immensely useful in constructing a mathematical model that describes where something is. 

We specify the directions we are talking about by drawing two little arrows of unit length (with NO dimensions or units!) in our two perpendicular directions. We then multiply these by a (positive or negative) distance with a unit. This allows us to separate (in algebra) direction from quantity.

The positive x-direction is specified by an arrow called "$\hat{i}$" with a little hat over it to show that it is a unit vector — a vector with size 1 and no dimensions.  The positive y-direction is specified by an arrow called "$\hat{j}$" with a little hat over it.

To get a vector that has units we multiply $\hat{i}$ and $\hat{j}$ by whatever we want the vector to be — for example, a distance, if we want a position (or displacement) vector. So we write:

$$\overrightarrow{r} = x\hat{i} + y\hat{j}$$

  • The "r" with an arrow over it is a position vector.
  • The "$\hat{i}$" and $\hat{j}$" are unit vectors specifying positive x and y directions. They are unitless (despite being called "unit vectors").
  • The "x" and "y"  are called coordinates. They have units and may be positive or negative, with a negative sign telling you to reverse the direction of the unit vector it is associated with.

Coordinates and directions

If we want to specify a vector in 2D, we can specify the x and y positive directions (using the unit vectors $\hat{i}$ and $\hat{j}$) and the x and y coordinates of the point in space that is at the tip of our vector. But this is not the only possible way to describe the vector in this coordinate system. We can also describe the total distance and the direction by using an angle — as in the figure at the right. 

These  two "representations" of a vector are related and you can convert from one to the other with math. If the displacement from the origin in the x direction is $x$, and the displacement from the origin in the y direction is $y$, then the total distance from the origin, $r$, and the angle of the direction from the x axis, $\theta$, can be found by the Pythagorean theorem and trigonometry to be

$$r=\sqrt{x^2 + y^2}$$

$$\tan{\theta} = \frac{y}{x}$$ 

These can be inverted to give

$$x = r \cos{\theta}$$

$$y = r \sin{\theta}$$.

Warning notes:

  • If we are only moving in 1D the appropriate way to write a displacement is $\overrightarrow{r} = x\overrightarrow{i}$, but often, especially in the beginning of this class, the x-coordinate is sometimes used by itself.
  • Sometimes a pair of coordinates is simply written as (x,y) — and this pair of numbers written with parentheses around them is described as a "vector". This is OK only if you are never going to change which coordinate system you use, since the actual directions of x and y ($\hat{i}$ and $\hat{j}$) are hidden.

Vectors in 3D

Sometimes we'll need vectors in 3D. Then we'll add a third unit vector, "$\hat{k}$", pointing perpendicular to the x-y plane and corresponding to the z coordinate.

Sometimes when we're working with vectors in 3D, we'll need to indicate on a screen or paper that a vector points out of the screen or into it. When we need to do this we'll use this symbol for "pointing at you out of and perpendicular to the screen": $\odot$. Think of it as the arrowhead looking as if the point of the arrow is coming towards you. We'll use this symbol for "pointing away from you into and perpendicular to the screen": ⊗. Think of it as the tail feathers of the arrow looking as if the arrow is going away from you.

Joe Redish and Wolfgang Losert 9/2/2012



Article 309
Last Modified: April 9, 2019