Converting Between Vector and Coordinate Representations

This page delves deeper into the conversion process between vector and coordinate representations of plane equations.

Converting from vector to coordinate representation:

1. Start with the vector form: n · $X - A$ = 0
2. Expand using the distributive property
3. Rearrange terms to match the coordinate form

Example: Vector form: $5, -1, -3$ · $X - (2, 1, 3)$ = 0 Expanded: 5x₁ - x₂ - 3x₃ - $10 - 1 - 9$ = 0 Coordinate form: 5x₁ - x₂ - 3x₃ = 0

Converting from coordinate to vector representation:

1. Identify the normal vector from the coefficients of x₁, x₂, and x₃
2. Choose a point on the plane (often by setting two variables to 1 and solving for the third)
3. Construct the vector equation using the normal vector and chosen point

Highlight: When converting to vector form, any point satisfying the plane equation can be used as the reference point A.

The page also reviews the scalar product (dot product) of vectors, which is crucial for these conversions.

Definition: The scalar product of two vectors a and b is defined as a · b = a₁b₁ + a₂b₂ + a₃b₃

Converting from Vector Representation to Parametric Form

This page explains the process of converting a plane equation from vector representation to Parameterform (parametric form).

The key steps in this conversion are:

1. Start with the vector form of the plane equation: n · $X - A$ = 0
2. Convert to coordinate form: n₁x₁ + n₂x₂ + n₃x₃ = d
3. Choose three points on the plane
4. Use these points to construct the parametric form

Example: Given the plane 5x₁ - x₂ - 3x₃ = 0, three points are chosen: P₁(2, 3, x₃), P₂(4, 5, x₃), P₃(9, 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 (points where the plane intersects the coordinate axes)

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.

Normal Form and Hessian Normal Form of Planes

This page introduces the different representations of planes in 3D space, focusing on the Normalenform (normal form) and the Hessesche Normalform (Hessian normal form).

The various forms presented include:

1. Vector representation of the normal form
2. Coordinate representation of the normal form
3. Parametric form
4. Vector representation of the Hessian normal form
5. Coordinate representation of the Hessian normal form

Definition: The normal form of a plane equation uses the normal vector of the plane and a point on the plane to define its position in space.

Vocabulary:

• Normalenvektor: Normal vector
• Aufpunkt: Point on the plane
• Richtungsvektor: Direction vector

The page also provides a visual example of how to determine a plane using three points and the cross product of direction vectors.

Example: To find the normal vector $n$ of a plane, the cross product of two direction vectors is calculated: n = v₁ × v₂

The conversion process from vector representation to coordinate representation is briefly outlined, showing how to expand the dot product and rearrange terms.

Highlight: The general form of the plane equation in coordinate representation is given as: n₁x₁ + n₂x₂ + n₃x₃ + d = 0, where n₁, n₂, n₃ are components of the normal vector, and d is a constant.