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Chapter 9: Problem 2

T/F: When plotting a point with polar coordinate \(P(r, \theta), r\) must be positive.

Short Answer

Expert verified
False, \(r\) can be negative in polar coordinates.

Step by step solution

01

Understanding Polar Coordinates

In polar coordinates, a point is represented by \(P(r, \ heta)\). Here, \(r\) is the radial distance from the origin to the point, and \(\theta\) is the angular coordinate, which measures the angle from the positive x-axis.
02

Interpreting Radial Distance

Radial distance \(r\) can be either positive or negative. If \(r\) is positive, the point is located in the direction of the angle \(\theta\) from the origin. If \(r\) is negative, the point is located in the opposite direction of the angle \(\theta\).
03

Drawing Conclusions

Since \(r\) can be negative, a point with polar coordinates does not necessarily require \(r\) to be positive. Therefore, the statement that \(r\) must be positive is false.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Radial Distance
In polar coordinates, the radial distance, denoted as \(r\), plays a crucial role as it determines how far a point is from the origin. Unlike Cartesian coordinates, which utilize \(x\) and \(y\) axes to pinpoint a location, polar coordinates make use of this radial distance to specify the position in a circular manner. Imagine the radial distance as the radius of a circle, stretching out from the center of a wheel to a point on its rim.
  • The radial distance \(r\) tells you how far from the center, or the origin, a point lies.
  • It simply measures the length of a straight line from the origin to the point.
The radial distance can vary in magnitude and sometimes even sign, which means it doesn't always point in the initially assumed direction. Understanding this concept is key to mastering polar coordinates and their applications.
Angular Coordinate
The angular coordinate, denoted as \(\theta\), is the second component of polar coordinates. It specifies the angle formed between the positive x-axis and the line drawn from the origin to the point. This is similar to how you would measure angles in a circle, starting from the rightmost edge (the positive x-axis) and moving counter-clockwise.
  • \(\theta\) is measured in degrees or radians, depending on the context or your preference.
  • The angle provides direction, supplementing the radial distance to precisely locate the point.
Both the radial distance and the angular coordinate work together to describe the exact position of a point on a plane in a way that's often more intuitive for certain applications, such as navigation and rotation.
Negative Radial Distance
An interesting and perhaps counterintuitive aspect of polar coordinates is the concept of negative radial distance. When \(r\) is negative, it doesn't indicate a tangible distance less than zero. Instead, it means the point is located in the direction opposite to \(\theta\). Imagine it like drawing a vector in the opposite direction of where \(\theta\) points.
  • A negative \(r\) flips the point from the intended direction by 180 degrees, essentially facing backwards.
  • This concept allows for a more flexible representation of points in polar coordinates, especially when plotting graphs with more complex symmetries.
Knowing that \(r\) can be negative expands our understanding of coordinate systems, illustrating the adaptability and usefulness of polar coordinates in various mathematical contexts.

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Most popular questions from this chapter

Answer the questions involving arc length. Approximate the arc length of one petal of the rose curve \(r=\sin (3 \theta)\) with Simpson's Rule and \(n=4\).

Convert the rectangular equation to a polar equation. \(x^{2} y=1\)

Parametric equations for a curve are given. Find \(\frac{d^{2} y}{d x^{2}},\) then determine the intervals on which the graph of the curve is concave up/down. . \(x=e^{t / 10} \cos t, \quad y=e^{t / 10} \sin t\)

Find \(t=t_{0}\) where the graph of the given parametric equations is not smooth, then find \(\lim _{t \rightarrow t_{0}} \frac{d y}{d x}\). \(x=\cos ^{2} t, \quad y=1-\sin ^{2} t\)

Graph the polar function on the given interval. \(\theta=\pi / 6, \quad=1 \leq r \leq 2\)
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