User talk:Paul Wormer/scratchbook1

PD Image
Equation for plane. X is arbitary point in plane; $\scriptstyle {\vec {\mathbf {a} }}$ and $\scriptstyle {\hat {\mathbf {n} }}$ are collinear.

In analytic geometry several closely related algebraic equations are known for a plane in three-dimensional Euclidean space. One such equation is illustrated in the figure. Point P is an arbitrary point in the plane and O (the origin) is outside the plane. The point A in the plane is chosen such that vector

${\vec {d}}\equiv {\overrightarrow {OA}}$

is orthogonal to the plane. The collinear vector

${\vec {n}}_{0}\equiv {\frac {1}{d}}{\vec {d}}\quad {\hbox{with}}\quad d\equiv \left|{\vec {d}}\,\right|$

is a unit (length 1) vector normal (perpendicular) to the plane. Evidently d is the distance of O to the plane. The following relation holds for an arbitrary point P in the plane

$\left({\vec {r}}-{\vec {d}}\;\right)\cdot {\vec {n}}_{0}=0\quad {\hbox{with}}\quad {\vec {r}}\equiv {\overrightarrow {OP}}\quad {\hbox{and}}\quad {\vec {r}}-{\vec {d}}={\overrightarrow {AP}}.$

This equation for the plane can be rewritten in terms of coordinates with respect to a Cartesian frame with origin in O. Dropping arrows for component vectors (real triplets) that are written bold, we find

$\left(\mathbf {r} -\mathbf {d} \right)\cdot \mathbf {n} _{0}=0\Longleftrightarrow xa_{0}+yb_{0}+zc_{0}=d$

with

$\mathbf {d} =(a,\;b,\;c),\quad \mathbf {n} _{0}=(a_{0},\;b_{0},\;c_{0}),\quad \mathbf {r} =(x,\;y,\;z),$

and

$\mathbf {d} \cdot \mathbf {n} _{0}={\frac {1}{d}}\mathbf {d} \cdot \mathbf {d} =d={\sqrt {a^{2}+b^{2}+c^{2}}}.$

Conversely, given the following equation for a plane

$ax+by+cz=e,\,$

it is easy to derive the same equation. Write

$\mathbf {r} =(x,\;y,\;z),\quad \mathbf {f} =(a,\;b,\;c),\quad {\hbox{and}}\quad \mathbf {d} \equiv \left({\frac {e}{a^{2}+b^{2}+c^{2}}}\right)\mathbf {f} .$

It follows that

$\mathbf {f} \cdot \mathbf {r} =e=\mathbf {f} \cdot \mathbf {d} .$

Hence we find the same equation,

$\mathbf {f} \cdot (\mathbf {r} -\mathbf {d} )=0\;\Longrightarrow \;(\mathbf {r} -\mathbf {d} )\cdot \mathbf {n} _{0}=0\quad {\hbox{with}}\quad \mathbf {n} _{0}={\frac {1}{\sqrt {a^{2}+b^{2}+c^{2}}}}\mathbf {f}$

where f , d, and n₀ are collinear. The equation may also be written in the following mnemonically convenient form

$\mathbf {d} \cdot (\mathbf {r} -\mathbf {d} )=0,$

which is the equation for a plane through a point A perpendicular to ${\overrightarrow {OA}}$.