### 6.1 - Rectangular coordinates

### The cartesian plane

A **plane** is a flat 2-dimensional region such as the surface of a page.

To specify locations on the plane we can lay down two number lines or **axes**
perpendicularly on the plane as shown to the right. The horizontal axis is usually
called the *x* axis and the vertical axis is usually called the *y* axis.
The point where the axes cross is usually chosen to be where *x* = 0 and where *y* = 0
(this point is called the **origin**).
The origin is indicated by a circle in the center of the picture.
The axes divide the plane into four regions called **quadrants**, which are numbered
counterclockwise as shown.
This construction, a plane plus two perpendicular axes, is
called a **cartesian plane** (named after René Decartes, 1596 - 1650).

If we draw points on a cartesian plane or if we draw a
curve representing a function
on a cartesian plane then we say that we are **graphing or drawing a graph** of the points or of the function.

The study of geometric shapes such as lines, circles, triangles, etc. can be done
on the ordinary, blank plane or on the cartesian plane. When done on the ordinary plane, their study
is called **plane geometry** and when done on the cartesian plane, it is called **analytical geometry**.
Trigonometry, the study of triangles, is done partly using
plane geometry and partly using
analytical geometry.

Any point on the cartesian plane can be located by giving its horizontal distance to the
right of the origin (this is called its *x* coordinate) and its vertical distance above
the origin (this is called its *y* coordinate). For example the red dot in the
cartesian plane shown to the right has an *x* coordinate of 2 and a *y* coordinate of 1.
Sometimes we say that this point is at *x* = 2 and *y* = 1 but
most often we use the **ordered pair notation** (2, 1) to describe this point.
(Note that the number before the comma is always the *x* coordinate and the
number after the comma is always the *y* coordinate.)

This method of using an *x* value and a *y* value to locate a point in the
plane is called the **rectangular coordinate system**
(notice the dotted rectangle shown in the picture).
Another coordinate system in common use, for example for
complex numbers, is the
**polar coordinate system**; it uses a distance from the origin and a direction to
locate a point.

### Identifying points on graphs

We saw above that giving an *x* coordinate and a *y* coordinate locates a
point in the cartesian plane. But we can turn the logic around: we could say that
a point in the cartesian plane represents the simultaneous values of two quantities
*x* and *y*. This interpretation is very important in science and
technology where *x* and *y* can represent almost any quantities.

The following examples show how to identify points on progressively more complicated
graphs.

**Example 1:** This graph has several features beyond what was explained above:
- In this graph the horizontal axis is called
*p* and the vertical axis *q*.
Thus points on this graph represent values of *p* and *q*.

- The labels
*p* and *q* are sometimes written beside the axes rather than at their ends.

- Only quadrant 1 was drawn, presumably because there is nothing interesting
to show in the other quadrants.

- The horizontal position of point
*A* is half-way between
*p* = 2 and *p* = 3. This mean that *p* = 2.5.
Similarly, the vertical position of point *A* is half-way between
*q* = 1 and *q* = 2.
Thus point *A* is at *p* = 2.5 and *q* = 1.5.

(Actually the fact that *p* is midway between 2 and 3 means that *p* = 2.5 only because the
horizontal scale is **linear**, which means that a movement of, say 1 centimeter, in the
horizontal direction represents the same change in the horizontal quantity anywhere on
the graph. There are other types of graphs such as
logarithmic graphs where this is not true.)

- In ordered pair notation, point
*A* is at (2.5, 1.5). Be careful!
This ordered pair does not mean *x* = 2.5 and *y* = 1.5; rather it means
*p* = 2.5 and *q* = 1.5.

**Example 2:** The labels on the axes now have units.
The label *t* (ms) means that *t* is measured in milliseconds
and the label *d* (cm) means that *d* is measured in centimeters.
Thus point *B* is at *t* = 250 ms and *d* = 7.5 cm.
Graphs with units appear often in scientific or technical applications.

**Example 3: ** The quantities plotted horizontally and vertically are now
**algebraic expressions**.
Thus at point *C*, the quantity *mg* has a value of 3 Newtons and the quantity
*r*^{ 2} has a value of 12.5 square meters.

Graphs with expressions on the axes appear often in scientific or technical applications.
Their use makes it possible to transform a curve into a straight line, which is
much easier to analyze.