Professor Dave here, let’s go 3D. All throughout this math series, we’ve been
All throughout this math series, we've been working with the coordinate plane, which has perpendicular X and Y axes. But this represents
only two spatial dimensions. Sometimes we will want to do math involving the third spatial
dimension that we experience every day. For this reason, it will be useful to learn the
basics regarding three-dimensional coordinate systems.
Let’s start with the coordinate plane and its two axes. In order to generate a three-dimensional
space, we simply have to add a third axis, a Z axis, which is perpendicular to the plane
containing the X and Y axes. And with that, we’ve gone from a two-dimensional plane
to a three-dimensional space. Whereas a point on a plane can be described with an ordered
pair, the first number giving the X coordinate and the second number giving the Y coordinate,
in space, we will need an ordered triple, where three numbers give us the X, Y, and
Z coordinates of the point. Convention requires that the three axes must be oriented in a
particular way, and this way is given by the right-hand rule. This tells us that if you
have the three axes, but you want to determine which direction on each axis is positive or
negative, simply curl your hand around the Z-axis. If your fingers are curling from the
positive side of the X-axis and towards the positive side of the Y-axis, in counterclockwise
fashion, then your thumb is pointing in the positive direction of the Z-axis. With these
axes all set, there is a lot to identify in this new space. First, there are now three
coordinate planes instead of one. Those are the XY, XZ, and YZ planes, which each contain
the two axes listed. Then, instead of the coordinate plane, which had quadrants, space
is divided up into eight octants, with the first octant being the one where X, Y, and
Z values are all positive. So any point in space will have three coordinates.
It can be a little tricky to represent these three dimensions in the two dimensions of
a screen or piece of paper, but if we draw lines or even a box, it usually becomes possible
to visualize it. In addition, whereas assigning a constant value to one of these variables
would produce a line in the two-dimensional coordinate plane, doing so in three-dimensional
space will produce a plane. The intersection of two planes, so long as they are not parallel,
will give us a line, and there are a number of ways a line can be represented algebraically,
including with vectors, which we will get to in a moment. One thing we should be aware
of is that just the way we can find the distance between two points in the coordinate plane
by using the distance formula, we can do the same thing in three-dimensional space, and
the analogous formula for three dimensions will not be very surprising if you remember
the old one. If we have two points, P one and P two, both with X, Y, and Z coordinates,
we can draw a box whereby the two points are opposite corners of the box, and the diagonal
of the box connects them. The length of this line segment, or the distance between the
two points, will be given by this equation, with X two minus X one quantity squared, plus
Y two minus Y one quantity squared, plus Z two minus Z one quantity squared, all under
a square root. So all we have done to the original distance formula is added this Z
term. We can arrive at this formula simply by drawing triangles within the box, and applying
the Pythagorean Theorem. Let’s call these two points A and B. (P one P two) quantity
squared is equal to (P one B) quantity squared plus B P two quantity squared. (P one B) quantity
squared, in turn, is equal to (P one A) quantity squared plus AB quantity squared, so we can
make that substitution now. Well P one A is just the difference in the X coordinates,
AB is the difference in the Y coordinates, and B P two is the difference in the Z coordinates,
so let’s represent them that way, and take the square root of both sides. That’s how
we get this distance formula. Let’s apply the formula and get the distance between the
point two, negative one, seven, and the point one, negative three, five. All we have to
do is plug these numbers into the formula, evaluate each difference, square the resulting
values, add those numbers together, and take the square root. So three will be our answer.
Now that we know how to use three-dimensional coordinates, let’s check comprehension.