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.

Converting from vector equation to Cartesian form

Using the scalar product to find the normal to two independant vectors

Slightly more complex vectors

Interesting alternative using the intercepts of the axes

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.

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)