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.
