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Chapter 10: Problem 6

Plot the point with these polar coordinates. $$[-2,0]$$

Short Answer

Expert verified
The point has been plotted 2 units to the left from the origin along the negative direction of the x-axis.

Step by step solution

01

Understand Polar Coordinates

The values in the brackets represent the Magnitude and Angle respectively in the form [Magnitude, Angle]. In this case, the magnitude is -2 (which represents the distance from the origin), and the angle is 0 degrees.
02

Plot the angle

An angle of 0 degrees usually lies along the positive x-axis. However, due to our negative magnitude, the direction will be reversed. Thus, the negative direction of the x-axis will be considered.
03

Plot the magnitude

Plot the distance from the origin. The magnitude tells you how far to go from the origin, since it's -2 we need to go 2 units to the negative direction of x-axis. Place your point there.

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Key Concepts

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

Magnitude and Angle
When discussing polar coordinates, it's important to understand the core components: the magnitude and the angle. The magnitude refers to the distance from the origin to the point. While in a typical radius, it's a positive value, polar coordinates can express this as negative to indicate direction. The angle, on the other hand, specifies the direction from the positive x-axis around the origin, measured in degrees or radians.

By conveying both the distance and direction, polar coordinates provide a unique way to locate a point in the plane. In terms of plotting, first consider the angle to establish direction, then consider the magnitude to determine distance along this direction.
Negative Magnitude
Unlike traditional Cartesian coordinates, where distance is always positive, polar coordinates can have a negative magnitude. This signifies that you must move in the opposite direction of the specified angle.

For instance, in the coordinates \([-2,0]\), the angle of 0 degrees indicates a point on the positive x-axis. However, the negative magnitude reverses this direction. Thus, instead of moving 2 units to the right (as a positive magnitude would suggest), you move 2 units left on the x-axis.
  • Negative magnitude means moving in the opposite direction.
  • This concept allows points to be plotted on both sides of the origin.
Understanding this allows for more flexible interpretations and location plotting.
Plotting Points
Plotting points using polar coordinates involves both magnitude and direction. It combines angular movement and linear distance in a unique way.

To plot a point like \([-2,0]\), start by identifying the angle. An angle of 0 degrees indicates the point is along the positive x-axis. However, our magnitude is negative, suggesting movement in the opposite direction. Therefore, instead of moving 2 units right, we proceed 2 units left along the x-axis.
  • Identify the angle to find direction.
  • Move by the magnitude but in reverse if it's negative.
This method showcases the flexibility and strength of polar coordinates in mapping points across different quadrants and orientations.

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