position of a point numerically as a sequence of real numbers. In

three-dimensional space this requires a sequence of three real numbers.

Using this representation, we can describe relationship among many

points in space (say, points lying on some plane or on a surface of some

sphere) algebraically.

This approach is fully utilized in a subject of Analytical Geometry that is studied in universities.

__Cartesian Coordinates in Space__

*Cartesian*coordinates in space are similar to Cartesian coordinates on a plane and just need an extension to a third dimension.

In any system of coordinates, first of all, we need a point of origin

*- some fixed point in space relatively to which we describe a position of any other point.*

**O**Next let's define

*axes*(plural of

*axis*) of coordinates.

I think, it's more natural to define first an axis that differentiates

three-dimensional geometry from two-dimensional one on a plane. So,

let's choose a line going through the point of origin

*, call it*

**O***Z-axis*and choose a particular direction on this line as positive.

We can conditionally call this Z-axis a

*vertical*axis because it is usually drawn vertically on pictures and illustrations.

Consider now a plane going through the point of origin

*perpendicular to Z-axis. This plane will contain two other axes -*

**O***X-axis*and

*Y-axis*- and we will call this plane

*XY-plane*.

We can choose X-axis first within this plane as we wish, then Y-axis

would be a line within this same plane and perpendicular to X-axis.

Both X-axis and Y-axis require a positive direction. Traditionally, we

choose a direction of X-axis as we wish and then choose a direction of

Y-axis in such a way that, looking from the positive ray of Z-axis,

positive ray of X-axis can be rotated counterclockwise to coincide with a

positive direction of Y-axis.

Similar construction could be done if we start from XY-plane, choose

X-axis and Y-axis within it and construct Z-axis as a perpendicular to

XY-plane.

Alternatively, we can start with only X-axis, then choose Y-axis

perpendicular to it, thus defining XY-plane, and then construct Z-axis.

Similar to XY-plane, we can define XZ-plane that contains X-axis and Z-axis or YZ-plane that contains Y-axis and Z-axis.

All these methods lead to similar result of having three axes of coordinates, X-axis (denoted as

*), Y-axis (denoted as*

**x***and Z-axis (denoted as*

**y***), mutually perpendicular to each other.*

**z**Finally, we have to choose a unit of measurement to be able to convert

geometrical position of any point in space into three coordinates

relative to three axes defined above.

Assume now that we have a point

*somewhere in space,*

**A**where three axes of coordinates are defined as described above. Our task

is to convert a geometrical position of this point to a sequence of

three real numbers that uniquely represent it. In other words, we will

put into one-to-one correspondence a set of all points in

three-dimensional space to a set of all triplets of real numbers.

The simple way to do it is to draw three plains through this point

*, each perpendicular to a corresponding axis (a unique construction):*

**A**plain

*⊥axis*

**α***(let*

**x***∩*

**α***=*

**x***)*

**A**_{x}plain

*⊥axis*

**β***(let*

**y***∩*

**β***=*

**y***)*

**A**_{y}plain

*⊥axis*

**γ***(let*

**z***∩*

**γ***=*

**z***)*

**A**_{z}The above defines three projections of point

*onto each axis -*

**A***,*

**A**_{x}*and*

**A**_{y}*, uniquely defined by a chosen position of point*

**A**_{z}*.*

**A**Having a unit of measurement, we can associate the length of segments

*,*

**OA**_{x}*and*

**OA**_{y}

**OA**_{z}with real numbers - their length in chosen unit, and assign a sign to

this number - positive if corresponding projection point lies towards

the positive direction of a corresponding axis or negative in an

opposite case.

The above procedure associates a triplet of real numbers with a position of a point in three-dimensional space.

These three numbers have proper names:

signed length of segment

*is called*

**OA**_{x}*x*-coordinate or

*abscissa*;

signed length of segment

*is called*

**OA**_{y}*y*-coordinate or

*ordinate*;

signed length of segment

*is called*

**OA**_{z}*z*-coordinate or

*applicate*.

Let's do an inverse operation to associate a point with a triplet of real numbers.

Obviously, three real numbers can be easily converted into three points

on our three axes by constructing three segments along them -

*,*

**OA**_{x}*and*

**OA**_{y}*. Then we construct three planes through points*

**OA**_{z}*,*

**A**_{x}*and*

**A**_{y}*correspondingly perpendicular to axes they belong to.*

**A**_{z}These three planes intersect in one and only one point

*, which is the point associated with given triplet of real numbers.*

**A**It would be a useful exercise to prove that our relationship between

points in three-dimensional space and triplets of real numbers is indeed

one-to-one correspondence. Rigorous proof of this is very much related

with foundations of geometry and certain axioms that are beyond the

scope of this course. Intuitively, however, it is as obvious as the fact

that for each real number

*r*we can find a point

*on a straight line that is on a distance equal to this number (using some unit of measurement) from a certain fixed point*

**R***, that is*

**O***=*

**OR***r*(for positive

*r*point

*is on one side from point*

**R***, for negative*

**O***- on the other).*

**R**As an exercise, here are a few examples of using coordinates to express the geometrical objects and properties.

1.

__Origin of coordinates__:

Point

**O***(0,0,0)*.

In a form of equations, it's a system of three equations

*x = 0*

*y = 0*

*z = 0*

2.

__Equation that describes all points on XY-plane__:

For any point on XY-plane a plane perpendicular to Z-axis is that same XY-plane.

It intersects Z-axis at an origin of coordinates, so its Z-coordinate must be equal to zero.

So, the equation of XY-plane is

*z = 0*

3.

__Equation of an angle bisector between Y-axis and Z-axis__:

Since all points within YZ-plane are characterized by an equation

*x = 0*and within YZ-plane points lying on an angle bisector between Y-axis and Z-axis are characterized by a property

*y = z*, the final numerical representation of this line is a system of two linear equations

*x = 0*

*y = z*

4.

__Equation describing points on a sphere of a radius__:

*R*with a center at the origin of coordinatesThe square of a distance from the origin of coordinates to a point

*with coordinates*

**A***(x,y,z)*is, as follows from Pythagorean Theorem applied twice, equals to

*x²+y²+z²*. Therefore, an equation for a sphere of radius

*R*is

*x²+y²+z² = R²*

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