Three Dimensional Coordinate Systems (Level 1)
Calculus III is a course that uses all the concepts learned from calculus I and extends
it to the study of functions with two or more variables. In a nutshell calculus III is calculus
I in 3 dimensions! Nothing more nothing less. With that said this class will challenge your
spatial visualization skills and push them to the limit.
Before we dive in we need to start thinking 3 dimensionally and be familiar with the 3-dimensional
coordinate System. We first need to remember the basics of number lines. Back in Algebra
I and perhaps even before that, you learned that a number line is formed when a numerical
value is assigned to each point on a line. For example the following line represents
a number line and point A is located at -1, likewise point B is located at 1 and half
or written another way as 1.5. Notice that number lines are useful to describe the location
of a point relative to the origin which is usually assigned a value of zero. In other
words, a number line can be viewed as a coordinate system, specifically a one dimensional coordinate
system consisting of all the points on a straight line. Notice that this line is broken into
two regions or half's the first half contains positive numbers and the second half contains
negative numbers and for the most part you can only move along the line either infinitely
to the left or infinitely to the right. In this sense, a 1-dimensional coordinates system
is equivalent to a number line where every real number, whether rational or irrational,
has a unique location or position on the line. Conversely, every point on the line can be
interpreted as a number in an ordered continuum that includes the real numbers. We can mathematically
denote this 1-dimensioanl space that includes the set of all real numbers as follows . Using
the real number symbol. Now if we take this number line and introduce
an additional number line that is perpendicular to this number line we obtain the modern coordinate
system, also known as the Cartesian coordinate system or rectangular coordinate system. A
two dimensional coordinate system specifies each point uniquely in a plane also known
as the xy plane, as oppose to a 1 dimensional coordinate system which specifies each point
in a straight line. As a result, a point on a two dimensional coordinate system is described
by a pair of numerical coordinates also known as an ordered pair, each coordinate in the
order pair describes the location of the point along each number line which are commonly
known as coordinate axis, The value along the x-axis also known as the abscissa is called
the x-coordinate of the ordered pair and is denoted first when describing a point on the
plane. Likewise the value, along the y axis also known as the ordinate is called the y-coordinate
and is denoted second when describing a point together the x and y coordinate form an ordered
pair. Notice that a point is measured relative to
a common center where the two number lines intersect this common point of intersection
is called the origin and is usually labeled by using the letter O the ordered pair that
describes this point is written as follows (0,0), as an example the following 4 points
can be described as follows: point A is located at (2,2) point B is located at (-1,1) point
C is located at (-2,-3) and point D is located at (4,-3). As you can see each point forms
a rectangle when plotting their position hence the rectangular coordinate system. Also Notice
that a two dimensional coordinate system is broken up into 4 regions also known as quadrants,
they are usually labeled as follows quadrant I quadrant II quadrant III and quadrant IV,
in addition the sign for each coordinate of an ordered pair is dependent on which quadrant
the point is located. If the point is located in quadrant I both the x and y coordinate
will be positive, in quadrant II the x coordinate will be negative and the y coordinate will
be positive, in quadrant III both the x and the y coordinate will be negative, and finally
in quadrant IV the x-coordinate will be positive and the y-coordinate will be negative. In
a two dimensional coordinate system you are free to move along two lines, infinitely to
the left, infinitely to the right, infinitely up and infinitely down. As oppose to just
one line like that in a one dimensional coordinate system. Mathematically we can define all the
points in a two dimensional coordinate system by the Cartesian product R times R or R squared
where R squared represents the xy-plane and it consist of all the ordered pairs where
the values of x and y are elements of the real numbers.
Let's move along and talk about the Three Dimensional Coordinate System also known as
a three dimensional rectangular coordinate system. If we take the two dimensional Cartesian
axis and add an additional number line Perpendicular to both axis we obtain a three dimensional
coordinate system. The three dimensional coordinate system contains 3 coordinate axes that intersect
at a common point also called the origin, these include the x-axis, y-axis and z-axis. For the most
part the x and y axis are horizontal lines and the z axis is a vertical line. Just like
the previous coordinate systems, each axis contains positive values and negative values,
on the x axis the positive values are located here, and the negative values are located
on the opposite direction. On the y axis the positive values are located here, and the
negative values in the opposite direction, and finally on the Z axis the positive values
are located here and negative values in the opposite direction. A way for you to remember
where the direction of the z axis is located is by using the right hand rule, if you curl
the fingers of your right hand around the z-axis counter clock wise from the positive
x-axis to the positive y-axis, then your thumb points towards the positive direction of the
z-axis. In addition the three coordinate axes also determine 3 coordinate planes. The xy
plane is the plane that contains the x and y axes; the yz plane contains the y and z
axes, and the xz plane contains the x and z axes. Together the three coordinate planes
divide space into eight regions called octants. This is analogous to the quadrants of a two
dimensional coordinate system, in this case a three dimensional coordinate system divides
space into octants. The first octant is determined by the positive axes, if you are having a
hard time visualizing this just look at any bottom corner of a room, that's essentially a visual representation
of the first octant. The wall on your right is the yz plane the wall on your left is the
xz plane and the floor is the xy plane. In the same manner there are 7 other rooms 3
more on top and 4 more on the bottom. If you imagine this three dimensional coordinate
system representing a two story building there would be 4 rooms on top and 4 rooms on the
bottom. Each room sharing a common corner called the origin.
In a three dimensional coordinate system a point is located in space as oppose to a two
dimensional system where a point is located on a plane and a 1 dimensional system where
a point is located on a line. With that said, we represent a point by an ordered triple x, y, z
of real numbers and we call the numbers the x, y and z coordinates of the point. In order
to locate a point in space we first start at the origin and move along the x axis to
locate the first coordinate then move parallel to the y axis for the second coordinate and
then parallel to the z axis to locate the last coordinate. The reason why this coordinate
system is called the rectangular coordinate system is because points in space determine
rectangular boxes. For example the coordinate (4,5,3) is plotted as follows, we first move along
the x axis 4 units in the positive direction then we move 5 units parallel to the y axis
in the positive direction, and then move parallel along the z axis also in the positive direction.
Now at first it's a little hard to see where the point is located so we need to draw a
corresponding box so we can get a better perspective and visualize this 3 dimensional plot on a
2 dimensional sheet of paper. The way we draw the box is actually pretty easy all we need
to do is draw 3 additional lines with the same length for each coordinate. So here we
draw 2 lines parallel to the first coordinate that starts from the point (0,5,0) and ends
at (4,5,0) another one that starts from (0,5,3) and ends at (4,5,3) and lastly one that starts
from (0, 0, 3) amd ends at (4,0,3). Then we draw lines parallel to the y coordinate one
for the bottom and two for the top of the box, then we do the same for the z coordinate
we draw lines parallel to connect the bottom and the top of the box. In the same way, the following
points would be plotted as follows: point B would be located at (4,-6,6)
point C would be located at: (-3,-3,2)
point D would be located at: (-6,3,4)
point E would be located at: (4,5,-5)
point F would be located at: (4,-6,-5)
point G would be located at: (-3,-3,-2)
and point H would be located at: (-6,3,-4)
This is how the points would look like in a sheet of paper, lets actually plot these points using a 3D plotter
Point A would look like this
Point B
Point C
Point D
Point E
Point F
Point G
and finally point H
so we graph this Cartesian plane and move it around like this
we see that we have 8 distinct points
four off them on top of the xy plane
and four of them on the bottom of the xy plane
one for each octant
it is really hard to see this by just using a two dimensional sheet of paper
but by using technology we can actually see a better representation of how these points look like in three dimensional space.
Just like the quadrants of a two dimensional coordinate system determines the sign of each
coordinate so does each octant, on the first Octant all 3 coordinates are positive, in
the second octant the x axis negative and both the y and z axis are positive, in the
third octant both x and y are negative and z is positive, in the 4th octant both x and
z are + and y is -, in the 5th octant both x and y are + and z is -, in the 6th octant x and
z are - and y is +, in the 7th octant, all 3 are negative, and finally in the 8th octant x
is + and both y and z are -. In a three dimensional coordinate system you
are free to move along three lines, infinitely to the west, infinitely to the east, infinitely
north, infinitely south, infinitely up and infinitely down. Mathematically we can define
all the points in a three dimensional coordinate system by the Cartesian product R times R
times R or R cubed where R cubed in this case 3d space consist of all the ordered triples
where the values of x y and z are elements of the real numbers.
Alright make sure you are comfortable with this new 3 dimensional coordinate system
it's going to be key in understanding the rest of calculus III, in our next video we
will focus on graphing equations using this new coordinate system.