How to Convert a Plane Equation from Coordinate Form to Parametric Form in 3 Steps
Introduction
In this article, we will learn how to convert a plane equation from coordinate form to parametric form in just three simple steps. This is a useful skill to have in mathematics, especially when dealing with geometry and linear algebra.
Step 1: Solve for a Variable
First, we need to solve the plane equation for one of the variables. Let's say we have the plane equation in coordinate form: ``` ax + by + cz = d ``` We can solve for x by subtracting by from both sides and then dividing by a: ``` x = (-by - cz + d) / a ```
Step 2: Substitute into Parametric Form
The parametric form of a plane equation is given by: ``` x = x0 + at y = y0 + bt z = z0 + ct ``` where (x0, y0, z0) is a point on the plane and (a, b, c) is a vector parallel to the plane. We can substitute our expression for x from Step 1 into the parametric form equation: ``` x = (-by0 - cz0 + d) / a + at ``` We can do the same for y and z: ``` y = y0 + bt z = z0 + ct ```
Step 3: Simplify
Finally, we can simplify our equations by factoring out the constants: ``` x = (d - by0 - cz0) / a + at y = y0 + bt z = z0 + ct ``` This is the parametric form of the plane equation.
Example
Let's say we have the plane equation in coordinate form: ``` 2x + 3y - 4z = 12 ``` We can follow the three steps above to convert it to parametric form: 1. Solve for x: ``` x = (12 - 3y + 4z) / 2 ``` 2. Substitute into parametric form: ``` x = (12 - 3y0 + 4z0) / 2 + at y = y0 + bt z = z0 + ct ``` 3. Simplify: ``` x = 6 - 3y0/2 + 2z0 + at y = y0 + bt z = z0 + ct ``` Therefore, the parametric form of the plane equation is: ``` x = 6 - 3y0/2 + 2z0 + at y = y0 + bt z = z0 + ct ```
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