key term - Radial Distance

Definition

Radial distance is the distance between the origin (0, 0) and a point on a polar coordinate plane. It represents the length of the line segment that connects the origin to the given point and is measured in the radial direction from the origin outward.

5 Must Know Facts For Your Next Test

1. Radial distance is represented by the variable $r$ in the polar coordinate system, where a point is denoted as $(r, \theta)$.
2. The radial distance is always a positive real number, as it represents a physical distance from the origin.
3. Radial distance is a key component in the conversion between Cartesian and polar coordinates, as $x = r\cos(\theta)$ and $y = r\sin(\theta)$.
4. The radial distance can be used to describe the size or magnitude of a vector in the polar coordinate system.
5. Plotting points in the polar coordinate system requires considering both the radial distance and the polar angle to accurately represent the location of the point.

Review Questions

• Explain the relationship between radial distance and the polar coordinate system.
  • Radial distance is a fundamental component of the polar coordinate system, which specifies the location of a point by its distance from the origin (radial distance, $r$) and the angle from the polar axis (polar angle, $\theta$). The radial distance represents the length of the line segment connecting the origin to the given point, measured outward from the origin. This distance is a crucial factor in converting between Cartesian and polar coordinates, as well as in describing the magnitude of vectors in the polar coordinate plane.
• Describe how radial distance is used to plot points in the polar coordinate system.
  • To plot a point in the polar coordinate system, you must consider both the radial distance ($r$) and the polar angle ($\theta$). The radial distance determines the length of the line segment from the origin to the point, while the polar angle specifies the direction of the point relative to the polar axis. By combining these two values, you can accurately represent the location of the point on the polar coordinate plane. The radial distance is a key factor in ensuring the point is placed at the correct distance from the origin.
• Analyze the significance of radial distance in the context of vector magnitudes and polar coordinate geometry.
  • Radial distance plays a vital role in the polar coordinate system, as it is used to describe the magnitude or size of a vector. The radial distance represents the length of the line segment connecting the origin to a given point, which corresponds to the magnitude of the vector pointing from the origin to that point. This relationship between radial distance and vector magnitude is crucial in understanding and working with vectors in the polar coordinate plane. Additionally, the radial distance, along with the polar angle, is essential for accurately plotting and analyzing the geometry of shapes and curves described in the polar coordinate system.

Related terms

Polar Coordinates:

A coordinate system that specifies the location of a point by its distance from a fixed reference point (the origin) and the angle from a fixed reference direction (the polar axis).

Polar Axis:

The reference direction from which the angle in a polar coordinate system is measured, typically the positive x-axis.

Polar Angle:

The angle between the positive polar axis and the line segment connecting the origin to the point, measured in a counterclockwise direction.