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Chapter 6: Problem 126

In the rectangular coordinate system, each point \((x, y)\) has a unique representation. Explain why this is not true for a point \((r, \theta)\) in the polar coordinate system.

Short Answer

Expert verified
A point in the polar coordinate system doesn't have a unique representation because the angle 胃 used to specify the position of the point can exceed 360掳 or be negative. Also, a point can be represented with a negative radius which means the direction of the point is in the opposite direction from the specified angle.

Step by step solution

01

Understanding the Coordinate Systems

In a rectangular coordinate system, each point is uniquely determined by an ordered pair of numbers: the x-coordinate, which specifies the horizontal position, and the y-coordinate, which defines the vertical position. In contrast, a point in a polar coordinate system is given by the distance (r) from the origin, called the radial coordinate or radius, and the angle (胃) from the positive x-axis, called the angular coordinate, polar angle, or azimuth.
02

Why Points in Rectangular Coordinates are Unique

In a rectangular coordinate system, there's only one way to specify a particular location, based on its x and y coordinates. This means you can't arrive at the same point with a different pair of coordinates. Therefore, each point (x, y) has a unique representation.
03

Why Points in Polar Coordinates are not Unique

In a polar coordinate system, more than one set of polar coordinates can represent the same point. This is because when specifying the position of a point, the angle 胃 can exceed 360掳 or be negative. Additionally, if r is negative, it means the point is in the direction opposite to the specified angle. As an example, the polar coordinates (2, 45掳) and (2, 405掳), and (-2, 225掳) would all map to the same point. Thus, in the polar coordinate system, a point does not have a unique representation

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