Introduction to the Cartesian Plane
The Cartesian plane, also called the rectangular coordinate plane or rectangular coordinates, is defined by two number lines that intersect at a right angle. This makes it possible to pictorially render equations that relate one variable to another. You should have a knowledge of middle-school algebra before tackling this chapter. Upon casual observation, some of the equations in this chapter look a little complicated, but nothing here goes beyond middle-school algebra.
Figure 6-1 illustrates the simplest possible set of rectangular coordinates. Both number lines have equal increments. This means that on either axis, points corresponding to consecutive integers are the same distance apart, no matter where on the axis we look. The two number lines intersect at their zero points. The horizontal (right-and-left) axis is called the x axis; the vertical (up-and-down) axis is called the y axis .
Fig. 6-1 . The Cartesian plane is defined by two number lines that intersect at right angles.
Ordered Pairs As Points
Figure 6-2 shows two specific points, called P and Q , plotted on the Cartesian plane. The coordinates of point P are (−5, −4), and the coordinates of point Q are (3,5). Any given point on the plane can be denoted as an ordered pair in the form (x,y) , determined by the numerical values at which perpendiculars from the point intersect the x and y axes. In Fig. 6-2, the perpendiculars are shown as horizontal and vertical dashed lines. When denoting an ordered pair, it is customary to place the two numbers or variables together right up against the comma. There is no space after the comma.
Fig. 6-2 . Two points P and Q , plotted in rectangular coordinates, and a third point R , important in finding the distance d between P and Q.
The word “ordered” means that the order in which the numbers are listed is important. The ordered pair (7,2) is not the same as the ordered pair (2,7), even though both pairs contain the same two numbers. In this respect, ordered pairs are different than mere sets of numbers. Think of a highway, which consists of a northbound lane and a southbound lane. If there is never any traffic on the highway, it doesn’t matter which lane (the one on the eastern side or the one on the western side) is called “northbound” and which is called “southbound.” But when there are cars and trucks on that road, it makes a big difference! The untraveled road is like a set; the traveled road is like an ordered pair.
Abscissa, Ordinate, And Origin
In any graphing scheme, there is at least one independent variable and at least one dependent variable . As the name suggests, the value of the independent variable does not “depend” on anything; it just “happens.” The value of the dependent variable is affected by the value of the independent variable.
The independent-variable coordinate (usually x ) of a point on the Cartesian plane is called the abscissa , and the dependent-variable coordinate (usually y) is called the ordinate . The point (0,0) is called the origin. In Fig. 6-2, point P has an abscissa of −5 and an ordinate of −4, and point Q has an abscissa of 3 and an ordinate of 5.
Fig. 6-2 . Two points P and Q , plotted in rectangular coordinates, and a third point R , important in finding the distance d between P and Q.
Distance Between Points
Suppose there are two different points P = ( x _{0} , y _{0} ) and Q = ( x _{1} , y _{1} ) on the Cartesian plane. The distance d between these two points can be found by determining the length of the hypotenuse, or longest side, of a right triangle PQR , where point R is the intersection of a “horizontal” line through P and a “vertical” line through Q . In this case, “horizontal” means “parallel to the x axis,” and “vertical” means “parallel to the y axis.” An example is shown in Fig. 6-2. Alternatively, we can use a “horizontal” line through Q and a “vertical” line through P to get the point R . The resulting right triangle in this case has the same hypotenuse, line segment PQ , as the triangle determined the other way.
Fig. 6-2 . Two points P and Q , plotted in rectangular coordinates, and a third point R , important in finding the distance d between P and Q.
Recall the Pythagorean theorem. It states that the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. In this case, that means:
d ^{2} = ( x _{1} − x _{0} ) ^{2} + ( y _{1} − y _{0} ) ^{2}
and therefore:
d = [( x _{1} − x _{0} ) ^{2} + ( y _{1} − y _{0} ) ^{2} ] ^{1/2}
where the 1/2 power is the square root. In the situation shown in Fig. 6-2, the distance d between points P = ( x _{0} , y _{0} ) = (−5, −4) and Q = ( x _{1} , y _{1} ) = (3,5) is:
d ={[3 −(−5)] ^{2} + [5−(−4)] ^{2} } ^{1/2}
= [(3 + 5) ^{2} + (5 + 4) ^{2} ] ^{1/2}
= (8 ^{2} + 9 ^{2} ) ^{1/2}
= (64 + 81) ^{1/2}
= 145 ^{1/2}
= 12.04(approx.)
This is accurate to two decimal places, as determined using a standard digital calculator that can find square roots.
The Cartesian Plane Practice Problems
PROBLEM 1
Plot the following points on the Cartesian plane: (−2,3), (3,−1), (0,5), and (−3,−3).
SOLUTION 1
These points are shown in Fig. 6-4. The dashed lines are perpendiculars, dropped to the axes to show the x and y coordinates of each point. (The dashed lines are not parts of the coordinates themselves.)
Fig. 6-4 . Illustration for Problems 1 and 2.
PROBLEM 2
What is the distance between (0,5) and (−3, −3) in Fig. 6-4? Express the answer to three decimal places.
SOLUTION 2
Use the distance formula. Let ( x _{0} , y _{0} ) = (0,5) and ( x _{1} , y _{1} ) = (−3, −3). Then:
d = [( x _{1} − x _{0} ) ^{2} + ( y _{1} − y _{0} ) ^{2} ] ^{1/2}
= [(−3 −0) ^{2} + (−3 − 5) ^{2} ] ^{1/2}
= [(−3) ^{2} + (−8) ^{2} ] ^{1/2}
= (9 + 64) ^{1/2}
= 73 ^{1/2}
= 8.544 (approx.)
Practice problems for these concepts can be found at: The Cartesian Plane Practice Test.
View Full Article
From Geometry Demystified: A Self-Teaching Guide. Copyright © 2003 by The McGraw-Hill Companies, Inc. All Rights Reserved.