Converting from Vector Representation to Parametric Form
This page explains the process of converting a plane equation from vector representation to Parameterform parametricform.
The key steps in this conversion are:
- Start with the vector form of the plane equation: n · X−A = 0
- Convert to coordinate form: n₁x₁ + n₂x₂ + n₃x₃ = d
- Choose three points on the plane
- Use these points to construct the parametric form
Example:
Given the plane 5x₁ - x₂ - 3x₃ = 0, three points are chosen:
P₁2,3,x3, P₂4,5,x3, P₃9,4,x3
The x₃ coordinate for each point is calculated by substituting into the plane equation.
Highlight: The parametric form of a plane is expressed as X = A + λAB + μAC, where A, B, and C are three non-collinear points on the plane, and λ and μ are parameters.
The process of finding the third coordinate x3 for each point involves substituting the known x₁ and x₂ values into the plane equation and solving for x₃.
Vocabulary:
- Parametergleichung: Parametric equation
- Spurpunkte: Trace points pointswheretheplaneintersectsthecoordinateaxes
The final step involves using the three calculated points to construct the parametric form of the plane equation.
Definition: The parametric form represents every point on the plane as a linear combination of two direction vectors, starting from a reference point.
This method provides a practical approach to converting between different representations of plane equations, which is crucial in many areas of mathematics and its applications.
