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Chapter 7: Problem 12

Plot the points, given in polar coordinates, on a polar grid. $$(1, \pi)$$

Short Answer

Expert verified
Plot the point at distance 1 from the origin and at an angle of 蟺 from the positive x-axis.

Step by step solution

01

Unpacking the polar coordinates

The polar coordinate given is (1, 蟺). The number 1 signifies the distance from the origin (r), and 蟺 is the angle formed with the positive x-axis in the counterclockwise direction (胃).
02

Plotting the point on a polar grid

On a polar grid, start from the origin and make a line segment of length 1 (r value) at an angle 蟺 from the positive x-axis in the counterclockwise direction. The point reached is the representation of the coordinate (1, 蟺) in polar terms.

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Key Concepts

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

Understanding the Polar Grid
A polar grid is a type of graphing system that differs from the typical Cartesian grid most people are familiar with. Instead of using a pair of perpendicular lines (axes) with a scale of units, a polar grid uses concentric circles and rays emanating from a central point known as the origin.
  • The circles, which are called "rings," indicate different radial distances from the origin.
  • The lines emanating from the center mark angles, generally measured in radians, from a standard position, typically the positive x-axis or "polar axis."
This layout allows you to plot points using two parameters: radius \(r\) and angle \(\theta\). Understanding how to navigate this grid is crucial when dealing with polar coordinates.
Plotting Points in Polar Coordinates
When changing from Cartesian coordinates to polar coordinates, one must think in terms of direction and distance from a central point. Each polar coordinate point is defined using two values, \(r \, \text{and}\, \theta\), where:
  • \(r\) indicates how far out from the origin the point lies.
  • \(\theta\) describes the angle from the positive x-axis measured counterclockwise.
To plot a point such as \( (1, \pi)\) in polar coordinates:
  • Start from the origin on a polar grid.
  • Move outward to 1 unit along the ring, as indicated by the radius value.
  • Then, rotate to an angle of \( \pi \) radians, or 180掳, to determine the direction.
The point on the grid now represents the coordinates given, succinctly enhancing your spatial understanding in a polar system.
Exploring Coordinate Systems
Coordinate systems are fundamental to graphing and mathematics because they provide a framework to locate a point on a plane. While the Cartesian system uses perpendicular axes to specify locations with \(x\) and \(y\) coordinates, polar coordinates provide an alternative by using radial distance \(r\) and angular displacement \(\theta\).
  • Cartesian coordinates are ideal for plotting linear relationships and easily grasping vertical and horizontal directions.
  • Polar coordinates simplify plotting of circular patterns and oscillations, providing an intuitive method where angles and radial distance play a primary role.
Both methods have unique advantages and applications, with polar coordinates being particularly useful in scenarios involving rotation, waves, and other phenomena best described in a circular context. Understanding these systems enhances your mathematical toolset, enabling you to choose the most efficient one for different types of problems.

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Most popular questions from this chapter

Round your answers to two decimal places. A golf ball is hit from a tee with a launch angle of \(13.2^{\circ}\) and speed 140 miles per hour. Express the velocity of the ball in component form. (Source: www.golf.com)

Find \(\mathbf{u}-\mathbf{v}, \mathbf{u}+2 \mathbf{v},\) and \(-3 \mathbf{u}+\mathbf{v}\). $$\mathbf{u}=\langle 6,-2\rangle, \mathbf{v}=\langle 3,-1\rangle$$

If \(u\) is a nonzero vector, for what values of \(k\) does the equation \(\|k \mathbf{u}\|=k\|\mathbf{u}\|\) hold? Explain.

Find the components of the vector in standard position that satisfy the given conditions. Magnitude \(10 ;\) direction \(190^{\circ}\)

This set of exercises will draw on the ideas presented in this section and your general math background. Explain why you cannot use the Law of Sines to solve an oblique triangle if you are given only the three sides of the triangle (SSS) and no two of them are of equal length.
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