Final answer:
To convert a cylindrical equation to rectangular coordinates, you replace the cylindrical terms with their rectangular counterparts: x = r cos(θ) and y = r sin(θ). For instance, a cylindrical equation 'r = 3' becomes 'x² + y² = 9' in rectangular coordinates.
Step-by-step explanation:
To find an equation in rectangular coordinates for a cylindrical equation, you need to express the cylindrical coordinates (r, θ, z) in terms of the rectangular coordinates (x, y, z). In the cylindrical coordinate system, r represents the distance of the point from the origin in the xy-plane, θ is the angle made with the positive x-axis, and z is the height above the plane.
In rectangular coordinates, the relationships are given by:
To express the given cylindrical equation, you'll need to replace r and θ with the corresponding expressions involving x and y.
For example, if you have a cylindrical equation such as r = 3, you would convert it to rectangular coordinates by replacing r with the square root of (x² + y²), yielding the equation √(x² + y²) = 3, which simplifies to x² + y² = 9. This represents the equation of a cylinder with a radius of 3 centered on the z-axis in rectangular coordinates.