Distance Calculations Continued

This page continues with distance calculations, focusing on the Abstand Punkt Ebene (point-to-plane distance). The formula for this distance is presented:

d(P; E) = |n₁P₁ + n₂P₂ + n₃P₃ - d| / √$n₁² + n₂² + n₃²$

Where (n₁, n₂, n₃) is the normal vector of the plane, (P₁, P₂, P₃) are the coordinates of the point, and d is the constant term in the plane equation.

An example is provided to demonstrate the application of this formula.

The page then introduces the concept of Abstand Gerade Gerade (line-to-line distance) for skew lines. The method involves using the cross product of the direction vectors of the two lines and the vector between points on the lines.

Vocabulary: Skew lines are lines in 3D space that are not parallel and do not intersect.

Example: For lines g₁: x = (5,1,3) + t$1,2,-1$ and h: x = (0,3,1) + s(1,1,2), the distance calculation involves finding the magnitude of the cross product of their direction vectors divided by the magnitude of the cross product of their direction vectors and the vector between points on the lines.

The page concludes with a note that for parallel lines, the calculation is similar to the point-to-line distance method.

Advanced Distance Calculations

This final page covers the remaining distance calculations in 3D space. It begins with the Abstand Gerade Ebene (line-to-plane distance). When a line is parallel to a plane, the calculation method is similar to the point-to-plane distance, using any point on the line.

The page then discusses the Abstand Ebene Ebene (plane-to-plane distance) for parallel planes. This calculation is approached similarly to the point-to-plane distance, using any point on one of the planes.

Definition: Parallel planes are planes in 3D space that never intersect, maintaining a constant distance between them.

Highlight: The Abstand Ebene Gerade parallel (distance between a plane and a parallel line) and Abstand paralleler Ebenen berechnen (calculating the distance between parallel planes) are important applications of these concepts in 3D geometry.

The page includes diagrams illustrating these distance calculations, showing the geometric relationships between lines, planes, and points in 3D space. These visual aids help reinforce the concepts and methods presented throughout the document.

Distance Calculations in 3D Space

This page introduces methods for calculating distances between different geometric entities in three-dimensional space. It covers the Abstand Punkt Punkt (point-to-point distance), which is simply the length of the vector between two points. An example is provided for calculating the distance between points P$3,1,2,-4$ and Q$4,-1,1,3$.

The page then moves on to explain the process for finding the Abstand Punkt Gerade (point-to-line distance). This involves three main steps:

1. Determining an auxiliary plane in normal form that is perpendicular to the line and contains the given point.
2. Finding the intersection point of the line and the auxiliary plane.
3. Calculating the length of the vector from the intersection point to the given point.

A detailed example is provided to illustrate this method, demonstrating how to set up the auxiliary plane equation and solve for the intersection point.

Example: For a point P$-2,1,2$ and a line g defined by x = (2,1,0) + r$-3,1,2$, the auxiliary plane equation is derived as $x-(-2)$ · $-3,1,2$ = 0, which simplifies to -3x + y + 2z = -3.

Highlight: The Abstand Punkt Gerade Lotfußpunkt (perpendicular foot point) is a crucial concept in this calculation, as it represents the point on the line closest to the given point.