Coordinate Systems Flashcards
What are the three main types of coordinate systems used in mathematics and physics?
Cartesian, cylindrical, and spherical coordinate systems.
In a Cartesian coordinate system, how is a point defined?
A point is defined by an ordered pair (x, y) in 2D or an ordered triplet (x, y, z) in 3D.
True or False: In cylindrical coordinates, the position of a point is defined using radius, angle, and height.
True.
What is the relationship between Cartesian coordinates (x, y, z) and cylindrical coordinates (r, θ, z)?
x = r * cos(θ), y = r * sin(θ), z = z.
Fill in the blank: In spherical coordinates, a point is defined by the radius, polar angle, and _______.
azimuthal angle.
How do you convert spherical coordinates (ρ, θ, φ) to Cartesian coordinates?
x = ρ * sin(φ) * cos(θ), y = ρ * sin(φ) * sin(θ), z = ρ * cos(φ).
What does ‘r’ represent in cylindrical coordinates?
The radial distance from the origin to the projection of the point onto the xy-plane.
True or False: The angle θ in cylindrical coordinates is measured from the positive y-axis.
False. It is measured from the positive x-axis.
What is the primary use of coordinate systems in mathematics?
To provide a framework for describing the position of points in space.
What is the formula to convert Cartesian coordinates (x, y) to polar coordinates (r, θ)?
r = √(x² + y²), θ = arctan(y/x).
Fill in the blank: In spherical coordinates, the angle θ is measured in the _______ plane.
xy-plane.
What is the range of the polar angle φ in spherical coordinates?
0 to π.
What type of coordinate system is most suitable for problems involving rotational symmetry?
Cylindrical coordinate system.
True or False: In Cartesian coordinates, all axes are perpendicular to each other.
True.
What are the coordinates of the origin in Cartesian space?
(0, 0, 0).
