converting between the different equations of planes
vector (Parametergleichung) to normal equation
- find the normal to the two given direction vectors
- cross product
- or solving the system of equations
normal equation to vector equation (Parametergleichung)
- set z = 0 and x = 0 for example (y=0, x=0 or y=0,z=0)
- use
$\vec{n} \cdot \vec{u} = 0$and$\vec{n} \cdot \vec{v} = 0$to find 2 linearly independent direction vectors in the plane - use the position vector already given in the normal equation.
normal equation to coordinate equation
- expand the scalar product
- calculate the scalar product of the position vector and the normal vector
- write out the scalar product of \vec x and the normal in the coordinate form.
Lage von zwei Ebenen
- parallel - how do we know if two planes are parallel? When do they coincide?
- intersecting
$E_1 \cap E_2$/ determine the line of intersection- Cartesian / Cartesian
- vector equation / Cartesian
angle between two planes
line intersecting a plane ($g \cap E$)
angle of a line intersecting a plane
position of a point relative to a plane
- in the plane
- mirroring a point with respect to a plane
Spurgeraden von Ebenen (intersection of the plane with the xy-, yz- and zx-plane)
- xy-plane: set z=0
- yz-plane: set x=0
- zx-plane: set y=0
Ebenenscharen
equation of planes with a parameter –> family of planes