equations of planes

vector equation (parametrische Gleichung)

$$ E: \vec{x} = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix} + s \begin{pmatrix} u_1 \\ u_2 \\ u_3 \end{pmatrix} + t \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}$$

normal equation (Normalengleichung)

$$ E: [ \vec{x} - \vec{a} ] \cdot \vec{n} = 0$$

Cartesian equation (Koordinatengleichung)

$$ E: n_1 x + n_2 y + n_3 z = d$$

where $ \vec{n} = \begin{pmatrix} n_1 \\ n_2 \\ n_3 \end{pmatrix}$ is the normal vector to the plane E.

training - Klausur 2

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
• use $n \cdot u = 0$ and $n \cdot v = 0$ 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
• intersecting / determine the line of intersection

Spurgeraden von Ebenen

Ebenenscharen

Exercises in Cornelsen book

p 159 - 161

Test

p 166

Review

angles

• lines/line
• line / plane
• plane / plane

review distance

HNF (Hesse’sche Normalenform) with a unit normal vector (Einheitsnormalenvektor)

The distance of a point P to a plane can be calculated using the Hesse’sche Normalenform.

$$ d = \frac{\left| (\vec{p} - \vec{a} ) \cdot \vec{n}\right| }{|\vec{n}|}$$ where $\vec{p}$ is the position vector of a point P and $\vec{a}$ is the position vector of a point of the plane.

$$\frac{\vec{n} }{|\vec{n}|}$$ is the unit normal vector.

Also see the page on distance.

applications - distance

• point / plane
• line or plane / parallel plane
• 2 parallel lines
• 2 skew lines (review)

more exercises

p 188-190

Test p 194