Radius vector

Definition

A radius vector is a line segment that extends from the origin of a coordinate system to a point in space, representing the position of that point in terms of distance and direction. In polar coordinates, the radius vector connects the pole (origin) to a point defined by its angle and distance from the pole, providing a visual representation of the relationship between angular and radial distances.

5 Must Know Facts For Your Next Test

1. In polar coordinates, the position of a point is given by its radius vector represented as (r, θ), where r is the length of the radius vector and θ is the angle it makes with a reference direction.
2. The length of the radius vector can be computed using the Pythagorean theorem when converting between polar and Cartesian coordinates.
3. The area swept by the radius vector as it moves through an angle θ can be calculated using the formula A = 1/2 * r^2 * θ.
4. The concept of a radius vector is crucial for calculating lengths of curves and areas in polar coordinates, enabling more straightforward integration methods.
5. The radius vector changes in both length and direction as it describes points that vary in angle and distance, which is essential for understanding curves in polar systems.

Review Questions

• How does the radius vector relate to polar coordinates and what does it represent visually?
  • The radius vector in polar coordinates represents the position of a point by connecting the origin to that point. It visually demonstrates how far away and in what direction the point is located from the origin, represented by its distance (r) from the origin and angle (θ) from a reference direction. This relationship allows for an understanding of how points are defined in terms of both radial distance and angular displacement.
• Discuss how you would use a radius vector to calculate areas in polar coordinates.
  • To calculate areas in polar coordinates using a radius vector, you can use the formula A = 1/2 * r^2 * θ. Here, r represents the length of the radius vector at a particular angle θ. By integrating this expression over a specified interval, you can determine the area enclosed by the curve defined by those angles, utilizing the varying lengths of radius vectors that sweep through those angles to capture all parts of the area.
• Evaluate how changing parameters of the radius vector affects calculations for arc lengths and areas in polar coordinates.
  • Changing parameters like the length of the radius vector or its angle will significantly impact calculations for arc lengths and areas in polar coordinates. For instance, increasing r while keeping θ constant will increase both arc length and area since these are directly proportional to r. Similarly, adjusting θ will determine how much area is swept out or how long an arc becomes, demonstrating how sensitive these calculations are to changes in both radial distance and angular position.

Related terms

Polar Coordinates: A two-dimensional coordinate system where each point is determined by a distance from a reference point and an angle from a reference direction.

Angle: The measure of rotation required to bring one line or plane into coincidence with another, usually measured in degrees or radians.

Arc Length:

The distance along a curve or circular path, which can be calculated using the radius and the angle subtended at the center of the circle.