Writing the Equation '5x + 4y + 7z = 1' in Spherical Coordinates
Spherical coordinates are a powerful tool in mathematics, particularly in the realm of three-dimensional geometry. When dealing with equations that involve multiple variables, such as the equation '5x + 4y + 7z = 1', converting the equation to spherical coordinates can provide valuable insights and simplify the problem-solving process.
In this article, we will guide you through the step-by-step process of writing the equation '5x + 4y + 7z = 1' in spherical coordinates, a crucial skill for high school mathematics students.
Understanding Spherical Coordinates
Spherical coordinates are a three-dimensional coordinate system that uses three values to represent a point in space: the radial distance (r), the polar angle (θ), and the azimuthal angle (φ). These coordinates are often denoted as (r, θ, φ).
The relationship between the Cartesian coordinates (x, y, z) and the spherical coordinates (r, θ, φ) is as follows:
- x = r sin(θ) cos(φ)
- y = r sin(θ) sin(φ)
- z = r cos(θ)
Converting the Equation to Spherical Coordinates
To write the equation '5x + 4y + 7z = 1' in spherical coordinates, we need to substitute the Cartesian coordinates (x, y, z) with their corresponding spherical coordinate expressions.
- Substitute the Cartesian coordinates:
- x = r sin(θ) cos(φ)
- y = r sin(θ) sin(φ)
- z = r cos(θ)
- Substitute the values into the original equation:
- 5(r sin(θ) cos(φ)) + 4(r sin(θ) sin(φ)) + 7(r cos(θ)) = 1
- Simplify the equation:
- r(5 sin(θ) cos(φ) + 4 sin(θ) sin(φ) + 7 cos(θ)) = 1
The resulting equation in spherical coordinates is:
r(5 sin(θ) cos(φ) + 4 sin(θ) sin(φ) + 7 cos(θ)) = 1
This equation represents the same surface as the original Cartesian equation '5x + 4y + 7z = 1', but in the spherical coordinate system.
By expressing the equation in spherical coordinates, you can gain a better understanding of the three-dimensional shape and properties of the surface defined by the equation. This knowledge can be particularly useful in various fields, such as physics, engineering, and advanced mathematics.