plane - Maple Help

geom3d

 plane
 define a plane

 Calling Sequence plane(p, [A, v]) plane(p, [A, dseg]) plane(p, [dseg1, dseg2]) plane(p, [l1, l2]) plane(p, [A, B, C]) plane(p, [A, l1, l2]) plane(p, eqn, n)

Parameters

 p - the name of the plane A, B, C - points v - vector dseg,dseg1, dseg2 - directed line segments l1, l2 - lines eqn - algebraic representation of a line (i.e., a polynomial or an equation) n - (optional) list of three names representing the names of the axes

Description

 • A plane p can be defined as follows:
 – from three given points A, B, and C.
 – from a given point A and a vector of dimension 3 v or a directed segment dseg. The plane defined is the plane that passes through A and has v or dseg as one of its normal vectors.
 – from two directed line segments with the same tail dseg1 and dseg2.
 – from two given lines l1 and l2. If l1 and l2 are parallel or intersect each other, the plane defined is the one that contains both l1 and l2. And if l1 and l2 are skew lines, p is the plane that contains l1 and is parallel to l2.
 – from a point A and two lines l1 and l2. The plane defined is the one that passes through A and is parallel to two lines l1 and l2.
 – from its algebraic representation eqn, i.e., eqn is a polynomial or an equation. In case the third optional argument is not given, if names are assigned to the three environment variables _EnvXName, _EnvYName, and _EnvZName then these three names will be used as the names of the axes.  Otherwise, Maple will prompt the user to input the names of the axes.
 • To access the information relating to a plane p, use the following function calls:

 form(p) returns the form of the geometric object (i.e., plane3d if p is a line). NormalVector(p) returns a normal vector of p which is a vector perpendicular to the plane p. Equation(p) returns the equation that represents the plane p. xname(p), yname(p), or zname(p) returns the name of the x-axis, y-axis, z-axis or FAIL if the axis is not assigned to any name. detail(l) returns a detailed description of the plane p.

 • The command with(geom3d,plane) allows the use of the abbreviated form of this command.

Examples

 > $\mathrm{with}\left(\mathrm{geom3d}\right):$

Find the equation of the plane through the origin parallel to each of the lines $\left(x-y+4z=1,2x+y-3z=2\right)$ and$\left(x+3=2y+1=3z+2\right)$.

Define the line $\left(x-y+4z=1,2x+y-3z=2\right)$

 > $\mathrm{plane}\left(\mathrm{p1},x-y+4z=1,\left[x,y,z\right]\right):$$\mathrm{plane}\left(\mathrm{p2},2x+y-3z=2,\left[x,y,z\right]\right):$
 > $\mathrm{line}\left(\mathrm{l1},\left[\mathrm{p1},\mathrm{p2}\right]\right):$

Define the line$\left(x+3=2y+1=3z+2\right)$

 > $\mathrm{line}\left(\mathrm{l2},\left[\mathrm{point}\left(A,-3,-\frac{1}{2},-\frac{2}{3}\right),\left[1,\frac{1}{2},\frac{1}{3}\right]\right]\right):$
 > $\mathrm{point}\left(o,0,0,0\right):$
 > $\mathrm{plane}\left(p,\left[o,\mathrm{l1},\mathrm{l2}\right]\right)$
 ${p}$ (1)
 > $\mathrm{detail}\left(p\right)$
 $\begin{array}{ll}{\text{name of the object}}& {p}\\ {\text{form of the object}}& {\mathrm{plane3d}}\\ {\text{equation of the plane}}& \frac{{13}{}{x}}{{6}}{+}\frac{{10}{}{y}}{{3}}{-}\frac{{23}{}{z}}{{2}}{=}{0}\end{array}$ (2)

Find the equation of the plane which bisects the join of $\left[\mathrm{x1},\mathrm{y1},\mathrm{z1}\right]$ and $\left[\mathrm{x2},\mathrm{y2},\mathrm{z2}\right]$ perpendicularly make sure the two given points are distinct

 > $\mathrm{assume}\left(\mathrm{x1}\ne \mathrm{x2}\right):$

Define two distinct points

 > $\mathrm{point}\left(A,\mathrm{x1},\mathrm{y1},\mathrm{z1}\right),\mathrm{point}\left(B,\mathrm{x2},\mathrm{y2},\mathrm{z2}\right):$
 > $\mathrm{midpoint}\left(C,A,B\right):$$\mathrm{line}\left(l,\left[A,B\right]\right):$
 > $v≔\mathrm{ParallelVector}\left(l\right):$

Define the plane which passes through the midpoint of $\mathrm{AB}$ and has $v$ as its normal vector

 > $\mathrm{plane}\left(p,\left[C,v\right]\right)$
 ${p}$ (3)
 > $\mathrm{Equation}\left(p,\left[x,y,z\right]\right)$
 ${x}{}\left({\mathrm{x2~}}{-}{\mathrm{x1~}}\right){+}{y}{}\left({\mathrm{y2}}{-}{\mathrm{y1}}\right){+}{z}{}\left({\mathrm{z2}}{-}{\mathrm{z1}}\right){-}\left({\mathrm{x2~}}{-}{\mathrm{x1~}}\right){}\left(\frac{{\mathrm{x1~}}}{{2}}{+}\frac{{\mathrm{x2~}}}{{2}}\right){-}\left({\mathrm{y2}}{-}{\mathrm{y1}}\right){}\left(\frac{{\mathrm{y1}}}{{2}}{+}\frac{{\mathrm{y2}}}{{2}}\right){-}\left({\mathrm{z2}}{-}{\mathrm{z1}}\right){}\left(\frac{{\mathrm{z1}}}{{2}}{+}\frac{{\mathrm{z2}}}{{2}}\right){=}{0}$ (4)