Parametrization of a plane

Vector calculus in plane and space: Straight Lines and Planes

Parametrization of a plane

Surfaces in the 3-dimensional space can also be described with the aid of parametric representations. Again, we choose a fixed origin $\vec{0}$ in the space. For a plane we need one particular vector and two direction vectors, and (thus) two parameters. To ensure that the parametric representation is truly a plane, the two direction vectors should not be on the same line through the origin.

Parameterization of a plane in the space

Let $\vec{u}$ and $\vec{v}$ be two vectors that are not on one line through $\vec{0}$ . The linear combinations $\lambda \cdot\vec{u}+\mu \cdot\vec{v}$ of $\vec{u}$ and $\vec{v}$ exactly pass through the points/vectors of a plane $U$ through the origin. We call this the plane spanned by $\vec{u}$ and $\vec{v}$ .

If $\vec{a}$ is a third vector, the point $\vec{x} = \vec{a}+\lambda \cdot\vec{u}+\mu \cdot\vec{v}$ passes through the plane $V$ through $\vec{a}$ parallel to $U$ , for varying $\lambda$ and $\mu$ . We call $\vec{x} = \vec{a}+\lambda \cdot\vec{u}+\mu\cdot \vec{v}$ a parametric or vector representation of $V$ .

The vectors $\vec{u}$ and $\vec{v}$ are called the direction vectors of both $U$ and $V$ .

The vector $\vec{a}$ is a particular vector (or particular point) of the parametric representation of the plane $V$ . We also call $\lambda$ and $\mu$ parameters.

The plane $U$ is the special case of the general plane $V$ , where $\vec{a}=\vec{0}$ is a position vector.

If $\vec{u}=r\cdot\vec{v}$ for a scalar $r$ , the points $\vec{a}+\lambda\cdot \vec{u}+\mu\cdot \vec{v}=\vec{a}+\left(\lambda\cdot r+\mu\right)\cdot \vec{v}$ for varying $\lambda$ and $\mu$ form nothing than the line with particular point $\vec{a}$ and direction vector $\vec{v}$ .

Just like the rights, particular and direction vectors are not uniquely determined: A plane can be described in several ways with particular and direction vectors. Each vector in $V$ may be selected as a particular point. Each basis of $U$ can be chosen as pair of direction vectors for $V$ .

The plane $V$ with parametric representation $\vec{x}= \vec{a}+\lambda \cdot\vec{u}+\mu \cdot\vec{v}$ can also be described with the parametric representation $\vec{x}= \vec{a}+\rho \cdot(\vec{u}+\vec{v})+\sigma \cdot(\vec{u}-\vec{v}).$
This means that every vector in the form $\vec{a}+\lambda \cdot\vec{u}+\mu \cdot\vec{v}$ can also be written in the form $\vec{a}+\rho \cdot(\vec{u}+\vec{v})+\sigma \cdot(\vec{u}-\vec{v})$ and vice versa. This fact also follows from the equalities
$\begin{array}{rcl}\vec{a}+\lambda \cdot\vec{u}+\mu\cdot \vec{v} &=& \vec{a} +\frac{1}{2}(\lambda + \mu)\cdot (\vec{u}+\vec{v}) + \frac{1}{2}(\lambda - \mu )\cdot(\vec{u}-\vec{v})\\ \vec{a}+\rho\cdot (\vec{u}+\vec{v})+\sigma\cdot (\vec{u}-\vec{v})&=&\vec{a}+(\rho + \sigma)\cdot\vec{u} + (\rho - \sigma ) \cdot\vec{v}\end{array}$

Instead of particular vector we will also speak of particular point. After all, we are talking about a point in the space, the endpoint of the representative of the support vector whose starting point is in the origin.

The definition of parallel corresponds to the geometric one: the planes $U$ and $V$ do not intersect.

Draw the plane $V$ with parametric representation $\rv{1,-5,-1}+\lambda\cdot\rv{-3,1,-1}+\mu\cdot\rv{-3,3,2}$

The position vector $\rv{1,-5,-1}$ is drawn in blue. The directional vector $\rv{-3,1,-1}$ and $\rv{-3,3,2}$ are drawn in black, and the plane $V$ is drawn in grey.

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