In the Cartesian coordinate system , the two axes are perpendicular to each other. Consider a coordinate system in which the x -axis and y -axis are non collinear (that is, the axes do not lie along a straight line) and are not perpendicular to each other (see the accompanying figure). a. Describe how a point is represented in this coordinate system by an ordered pair ( x , y ) of real numbers. Conversely, show how an ordered pair ( x , y ) of real numbers uniquely determine a point in the plane. b. Suppose you want to find a formula for the distance between two points, P 1 ( x 1 , y 1 ) and P 2 ( x 2 , y 2 ) , in the plane. What advantage does the Cartesian coordinate system have over the coordinate system under consideration?
In the Cartesian coordinate system , the two axes are perpendicular to each other. Consider a coordinate system in which the x -axis and y -axis are non collinear (that is, the axes do not lie along a straight line) and are not perpendicular to each other (see the accompanying figure). a. Describe how a point is represented in this coordinate system by an ordered pair ( x , y ) of real numbers. Conversely, show how an ordered pair ( x , y ) of real numbers uniquely determine a point in the plane. b. Suppose you want to find a formula for the distance between two points, P 1 ( x 1 , y 1 ) and P 2 ( x 2 , y 2 ) , in the plane. What advantage does the Cartesian coordinate system have over the coordinate system under consideration?
Solution Summary: The author explains that the given statement is defined below. The distance between two points is unique in the Cartesian system but in non-collinear system is not unique.
In the Cartesian coordinate system, the two axes are perpendicular to each other. Consider a coordinate system in which the
x
-axis and
y
-axis are non collinear (that is, the axes do not lie along a straight line) and are not perpendicular to each other (see the accompanying figure).
a. Describe how a point is represented in this coordinate system by an ordered pair
(
x
,
y
)
of real numbers. Conversely, show how an ordered pair
(
x
,
y
)
of real numbers uniquely determine a point in the plane.
b. Suppose you want to find a formula for the distance between two points,
P
1
(
x
1
,
y
1
)
and
P
2
(
x
2
,
y
2
)
, in the plane. What advantage does the Cartesian coordinate system have over the coordinate system under consideration?
System that uses coordinates to uniquely determine the position of points. The most common coordinate system is the Cartesian system, where points are given by distance along a horizontal x-axis and vertical y-axis from the origin. A polar coordinate system locates a point by its direction relative to a reference direction and its distance from a given point. In three dimensions, it leads to cylindrical and spherical coordinates.
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