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.
