Cartesian Plane

Are you ready to delve into the realm of Cartesian planes? Well, you're in the right place! This lesson will introduce you to the ins and outs of this core geometric concept in a clear and straightforward manner.

At its most basic, a Cartesian plane is a geometric representation of a plane, grounded in two axes that intersect at right angles, meeting at a specific point known as the origin, or "O".

Drawing a Cartesian Plane

First things first, let's get our hands dirty.

Grab a pen and paper and draw a vertical line, marking the upward direction.

Then, do the same with a horizontal line, with the direction moving towards the right.

Keep in mind that these lines need to be orthogonal—that's just a fancy term for them being at right angles with each other—forming four crisp 90° angles.

The point where these lines intersect is the heart of our Cartesian plane: the origin (O).

 

Now, onto terminology: the horizontal axis is also known as the x-axis, or abscissa. The vertical line? That's the y-axis, or ordinate.

Next up, it's time to measure the Cartesian axes. Choose a unit of measure for this purpose.

 

Quick note: While it's possible to use different units for both axes, it's typically easier to stick with one unit of measure. For simplicity, let's use 1 cm for both the x and y axes.

The Cartesian plane is populated by points, each corresponding to ordered pairs of real numbers—our coordinates (x;y).

• The first member of the pair, x, is our abscissa, denoting a point on the horizontal x-axis.
• The second member of the pair, y, is our ordinate, indicating a point on the vertical y-axis.

To pinpoint a point's coordinates on the plane, imagine or draw a line (r) parallel to the y-axis that passes through the point P.

Where line r intersects the x-axis (at x=4, in this case), that's the abscissa of point P, or in simpler terms, x=4.

Let's keep going. Draw another line (s) parallel to the x-axis that also passes through point P.

Line s intersects the y-axis at y=3. And there we have it—the ordinate of point P, or y=3.

With the abscissa (x=4) and the ordinate (y=3) in hand, we've nailed down the coordinates of the point.

So, in our example, point P is hanging out at coordinates (4;3).

$$ P:(x;y)= (4;3) $$

The lines r and s are called the projections of point P onto the Cartesian axes. They cross paths at one, and only one, point on the plane with coordinates (x,y). This means every point on the plane corresponds to an ordered pair of real numbers (x,y), and vice versa.

An Example

Ready to put this into action? Draw up a Cartesian diagram.

Now, let's find point P at coordinates (4;3).

Sketch a line (r) perpendicular to the x-axis through the abscissa at x=4.

Then, sketch a line (s) perpendicular to the y-axis through the ordinate at y=3.

The intersection of lines r and s reveals point P on the Cartesian plane.

So, our point P is cozily situated at coordinates (4;3).

Dividing the Plane: Quadrants of the Cartesian Plane

Our Cartesian axes neatly split the plane into four quarters, known as quadrants.

Always label the quadrants counterclockwise, beginning with the first quadrant.

• The first quadrant is home to points with positive abscissa and ordinate (x>0 , y>0).
  
• The second quadrant houses points with a negative abscissa (x<0) and positive ordinate (y>0).
  
• The third quadrant is where you'll find points with both negative abscissa and ordinate (x<0 , y<0).
  
• And finally, the fourth quadrant is where points with a positive abscissa (x>0) and negative ordinate (y<0) reside.
  

One thing to note: points on the Cartesian axes and at the origin (O) aren't part of any quadrant—they are the boundary points.