Video: Coordinate Systems

Video Transcript

In this video, we’re going to talk about coordinate systems, what they are, why they’re useful, and how to make use of them. It happens many times that we ask questions of a quantitative nature, questions like how far, or in what direction, how fast is something moving, or how long does a process take. Each of these questions asks for a numerical answer, often based on measurement. And when we want a numerically measured value, we’ll often create a coordinate system in order to make that measurement. Let’s move ahead by talking a bit more about why coordinate systems are so useful.

First, what is a coordinate system? A coordinate system is a framework that we use for uniquely specifying the location of a point, typically on a grid. This is important because it means that all of our locations are arranged in an orderly fashion. Flowing out of this property, we can say that coordinate systems provide an organized way of collecting quantitative information.

Coordinate systems provide a basis and a reference for measuring quantitative information. And we use them all the time. Whenever we look to a thermometer to see what the outside temperature is, or whenever we’re driving along the road and we look at our speedometer to see how fast we’re going. In both cases, we’re making a quantitative measurement based on a coordinate system.

Now, you may be getting the idea that a coordinate system could be created out of almost any standard or reference point. And that’s right. That said, over time, some systems have proven to be more useful or more universally applicable than others. Let’s talk now a little bit about two of those more common coordinate systems.

The first system, and probably the most commonly found, is called the Cartesian coordinate system, named after Rene Descartes. This system consists of two orthogonal, or perpendicular, axes joined at a point we’ve called the origin. Measurements along one direction are independent of measurements along the other direction. So, if we want to specify the particular or unique location of a point 𝑃 on this grid, then we’ll need two bits of information. We’ll need the horizontal, or 𝑥-coordinate, of that point as well as the vertical, or 𝑦-coordinate. Once these two points are inserted, we’ve uniquely specified the location of a point in our grid.

The Cartesian coordinate system need not be limited to two dimensions. We can add a third dimension, often called the 𝑧-direction, which itself is perpendicular to the other two directions, 𝑥 and 𝑦. In this case, our point 𝑃 would then need a third bit of information in order to uniquely specify that location. We would need to include the 𝑧-axis coordinate.

The process we’ve used so far will let us specify the location of particular points in our grid. But what if we wanna specify not just a point, but a vector? That is, a particular distance and direction within our three-dimensional Cartesian system. In order to describe a vector like the vector 𝑃, we use what are called unit vectors.

Strictly speaking, 𝑥, 𝑦, and 𝑧 are not vectors. They’re coordinate values we can measure in order to uniquely specify points in our grid. But the 𝑥, 𝑦, and 𝑧 framework is quite useful to us because these dimensions are perpendicular to one another. We can supplement this strength by defining a vector, called a unit vector, for each direction that our coordinate system includes.

The 𝑥-dimension typically has a unit vector called 𝑖. The 𝑦-dimension has a unit vector called 𝑗. And the 𝑧-dimension has a unit vector we’ve called 𝑘. The name unit vector comes from the fact that the magnitude of each one of these three vectors is equal to one. Each one advances one unit along its given dimension.

Returning to our vector 𝑃, we’ve now gained the ability to describe this vector within the Cartesian coordinate system. We’ll do it by using unit vectors. The vector 𝑃, which joins the origin with the point two, three, and 𝑧 sub 𝑃, the vector 𝑃 points two units in the 𝑖-direction. That is along the 𝑥-axis. It points three units in the 𝑗-direction. That is along the 𝑦-axis. And it points 𝑧 sub 𝑃 units in the 𝑘-direction. That is along the 𝑧-axis. Using a Cartesian coordinate system, we now have the ability to describe points in three-dimensional space as well as describe the vectors created by joining the origin with those points.

Before moving on to our second coordinate system, let’s consider one last question about this point 𝑃. And the question is, how far is 𝑃 from the origin? That is, what is the magnitude, or the length, of the vector 𝑃? If we return to a two-dimensional Cartesian system, then if we wanna find the distance of a point from the origin. If we consider the right triangle that connecting this point with the origin and dropping a vertical line down from it to the 𝑥-axis makes.

If we call the lengths of the sides 𝐴 and 𝐵 and the length of the hypotenuse 𝐶, we may recall the Pythagorean theorem. Which tells us that the length of the hypotenuse 𝐶 squared is equal to 𝐴 squared plus 𝐵 squared. Or, 𝐶 equals the square root of 𝐴 squared plus 𝐵 squared. This relationship is the same relationship we’ll use to calculate distance either in two dimensions or, as in our case with point 𝑃, in three dimensions.

In general, for any vector 𝑉 in three-dimensional Cartesian space, its magnitude, or length, is equal to the square root of the change in its 𝑥-component squared plus the change in its 𝑦-component squared plus the change in its 𝑧-component squared. So, in the case of vector 𝑃, its magnitude, or length, would be the square root of two squared plus three squared plus 𝑧 sub 𝑃 squared. The Cartesian coordinate system is by far the most common system we’ll encounter. But there’s a second system it’s worth knowing a bit about too.

The polar coordinate system is a two-dimensional coordinate system where points are specified not by 𝑥 and 𝑦 but by 𝑟 and 𝜃. Given a point 𝑃 in our grid, we would start at what’s called the polar axis. We would rotate counterclockwise about that axis an angle 𝜃 and then move out from the origin a distance 𝑟, at which point we would uniquely specify and arrive at the point 𝑃.

The polar coordinate system is especially useful in systems where there is circular rotation. It’s possible to convert between the polar and Cartesian coordinate frames. Given the polar coordinates 𝑟 and 𝜃, the Cartesian coordinates 𝑥 and 𝑦 can be solved for, where 𝑥 equals 𝑟 times the cos of 𝜃 and 𝑦 is 𝑟 times the sin of 𝜃. Now that we’ve learned a bit about these two systems, let’s get some practice using them.

The coordinates of a particle in a rectangular coordinate system are 1.0, negative 4.0, and 6.0. What is the position vector of the particle?

Let’s call the position vector of the particle we’re looking for 𝑃. And if we write out 𝑃 as a point in space, that point is given as 1.0, negative 4.0, 6.0, where each number refers to a value along a dimension. We’re told we’re working in a rectangular coordinate system. This is another name for the Cartesian system.

We can draw a three-dimensional coordinate system where 𝑥, 𝑦, and 𝑧 specify those three perpendicular directions. The coordinates of our point 𝑃 all match up, each one with a particular direction, the first with 𝑥, the second with 𝑦, and the third with 𝑧. If we plot our point 𝑃 on this set of axes, we know its 𝑥-value will be 1, its 𝑦-value will be negative 4.0, and its 𝑧-value will be positive 6.0. If we find the line along which 𝑥 equals one and 𝑦 is equal to negative four, then if we go out 6.0 units in the positive 𝑧-direction along that line, then we’ll find where the point 𝑃 is located.

In this exercise, we wanna solve for the position vector of that point 𝑃. We can draw the position vector 𝑃 in on our diagram as starting at the origin and going to the point 𝑃. For each of our three dimensions, 𝑥, 𝑦, and 𝑧, there is a corresponding unit vector, 𝑖 for 𝑥, 𝑗 for 𝑦, and 𝑘 for 𝑧. These unit vectors are the building blocks that we’ll use to create vectors in this three-dimensional space. With this correlation known, we can now write out the vector 𝑃 by referring to the coordinates of the point 𝑃.

The vector 𝑃 will have 1.0 unit in the 𝑖-direction. In the 𝑦-dimension, it will go negative 4.0 units, or negative 4.0𝑗. And along the 𝑧-axis, it will move positive 6.0 units, or 6.0 in the 𝑘-direction. This is the vector that describes the position of the particle.

Now let’s try another example, this time, using the polar coordinate system.

Two points in a plane have polar coordinates 𝑃 sub one 2.500 meters, 𝜋 over six and 𝑃 sub two 3.800 meters, two 𝜋 over three. Determine the Cartesian coordinates of 𝑃 sub one. Determine the Cartesian coordinates of 𝑃 sub two. Determine the distance between the points, to the nearest centimeter.

We can call the Cartesian coordinates of point 𝑃 sub one 𝑥 one, 𝑦 one and the Cartesian coordinates of 𝑃 sub two 𝑥 two, 𝑦 two. The distance between these two points we’ll call capital 𝐷. As starting information in this exercise, we’re given two points, 𝑃 sub one and 𝑃 sub two, in their polar coordinate setup. This means that the first coordinate in each pair is the radial distance. We’ll call that 𝑟 sub one for 𝑃 sub one and 𝑟 sub two for 𝑃 sub two. The second coordinate in the pair is the angular coordinate. In 𝑃 sub one, we’ll call that value 𝜃 sub one. And in 𝑃 sub two, we’ll call it 𝜃 sub two.

We can recall the coordinate conversion relationships between polar coordinates and Cartesian or 𝑥-, 𝑦-, 𝑧-coordinates. In our two-dimensional setup, if we take the polar coordinate 𝑟 and multiply it by the cosine of the angular polar coordinate 𝜃, then we’ll get the Cartesian coordinate 𝑥. Similarly, if we take that coordinate 𝑟 and multiply it by the sine of the angle 𝜃, we’ll get the Cartesian coordinate 𝑦.

This means that the Cartesian coordinates of the first point 𝑥 one, 𝑦 one are equal to 𝑟 sub one cos 𝜃 sub one and 𝑟 sub one sin 𝜃 sub one. When we plug these values in, using 2.500 meters for 𝑟 sub one and 𝜋 over six for 𝜃 sub one, we find a result of 2.165 meters in the 𝑥-direction and 1.250 meters in the 𝑦. These are the Cartesian coordinates of the point 𝑃 sub one.

Likewise, for 𝑥 two, 𝑦 two, where we’ll use polar values 𝑟 sub two and 𝜃 sub two. Using a value of 3.800 meters for 𝑟 sub two and two 𝜋 over three for 𝜃 sub two, we find a result of negative 1.900 meters in the 𝑥-direction and 3.291 meters in the 𝑦-direction. These are the Cartesian coordinates of the polar point 𝑃 sub two.

Finally, we want to solve for the distance 𝐷 between these two points. That distance, mathematically, is equal to the square root of the change in 𝑥 squared plus the change in 𝑦 squared. In our case then, 𝐷 is equal to the square root of 𝑥 two minus 𝑥 one quantity squared plus 𝑦 two minus 𝑦 one quantity squared. When we plug in our values for 𝑥 two, 𝑥 one, 𝑦 two, and 𝑦 one and then enter this expression on our calculator, we find that 𝐷 is 4.55 meters. That’s the distance between these two points, to the nearest centimeter.

In summary, coordinate systems let us make measurements and compare differences quantitatively. Two of the most common coordinate systems we’ll see are Cartesian, where we use 𝑥-, 𝑦-, and 𝑧-dimensions to describe points, and polar, where we use 𝑟 and 𝜃 to describe point locations. And in a three-dimensional Cartesian system, the unit vectors 𝑖, 𝑗, and 𝑘 describe vectors in the 𝑥-, 𝑦-, and 𝑧-dimensions, respectively.