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Student[MultivariateCalculus]

 Plane
 Create and initialize a plane object

 Calling Sequence Plane(eqn, opts) Plane(expr, opts) Plane(p1, p2, p3, opts) Plane(p1, v1, opts) Plane(v1, p1, opts) Plane(p1, p2, v1, opts) Plane(p1, v1, p2, opts) Plane(v1, p1, p2, opts) Plane(p1, v1, v2, opts) Plane(v1, p1, v2, opts) Plane(v1, v2, p1, opts) Plane(p1, l1, opts) Plane(l1, p1, opts) Plane(l1, l2, opts) Plane(P, p1, opts) Plane(P, opts)

Parameters

 eqn - Linear equation defining the plane expr - Expression linear in the coordinate variables, equated to 0 to define the plane p1, p2, p3 - Points on the plane, expressed as lists of coordinates v1, v2 - Vector used to define the plane; either a normal Vector, or directions lying within the plane - see below for details l1, l2 - Line objects used to define the plane; either in the plane or parallel to the plane - see below for details P - Plane object defined earlier; the plane to be defined will be parallel (or equal) to this plane opts - (optional) equation defining the variables to be used in the representation

Options

 • The opts arguments can contain the following option.
 • variables = list
 The variables to be used in the plane's equation.  This is used in interpreting the equation and expression forms of the call to the Plane function, and when obtaining equations for the plane. The default is $\left[x,y,z\right]$.
 • id = positive integer, name, or string
 Planes display as $\mathrm{<< Plane n >>}$, where $n$ is an identification for the plane that is by default a positive integer assigned in order of creation. The id option can be used to force the plane to be given a different identification. It is an error to use the same identification for two different planes that are both in use.

Description

 • The Plane command creates a plane object that can be operated on or graphed.
 • Planes can be specified in the following ways:
 – A linear equation eqn, such as $ax+by+cz+d=0$.
 – A linear expression expr of the form $ax+by+cz+d$, which is set equal to 0 to obtain the plane.
 – Three points p1, p2, p3, such as $\left[1,2,3\right]$, $\left[1,0,-1\right]$, $\left[0,1,2\right]$, lying in the plane. The points must not lie on a line.
 – A point p1, such as $\left[1,2,3\right]$, and a Vector v1, such as $⟨2,0,-1⟩$. The plane contains p1 and has v1 as its normal. The arguments can also be specified in the other order.
 – Two points, p1 and p2, such as $\left[1,2,3\right]$ and $\left[1,0,-1\right]$, lying in the plane, and a Vector v1, such as $⟨1,0,2⟩$, the direction of which lies within the plane. The arguments can be specified in any order. The Vector v1 cannot be a multiple of the Vector from p1 to p2.
 – A point p1, such as $\left[1,2,3\right]$, and two Vectors v1 and v2, such as $⟨0,1,2⟩$ and $⟨1,0,2⟩$. The directions of both Vectors lie within the plane. The arguments can be specified in any order. The Vectors v1 and v2 must be linearly independent.
 – A point p1, such as $\left[1,2,3\right]$, lying in the plane, and a line l1 defined earlier, say $\mathrm{Line}\left(\left[0,0,0\right],\left[0,1,2\right]\right)$, also lying in the plane. The arguments can be specified in any order. The line l1 cannot contain the point p1.
 – Two Line objects l1 and l2 defined earlier, say $\mathrm{Line}\left(\left[1,2,3\right],\left[1,0,-1\right]\right)$ and $\mathrm{Line}\left(\left[0,0,0\right],\left[0,1,2\right]\right)$. If the lines intersect or are parallel, then this defines the plane containing both lines. If the lines are skew, then it defines the plane containing l1 and parallel to l2. If the lines are equal, an error is returned.
 – A Plane object P, defined earlier, such as $\mathrm{Plane}\left(x+y+z=0\right)$, and a point p1, such as $\left[1,2,3\right]$. This defines the plane through p1 that is parallel to P.
 – A plane object P, defined earlier, such as $\mathrm{Plane}\left(x+y+z=0\right)$. (This can be useful to change the names of the coordinate variables and the parameter, using the options explained below.)
 • It is possible to specify two mathematically identical planes using different Plane commands.
 • The coefficients and coordinates defining a Plane can contain parameters. For example, we can define a plane containing the points $\left[2,3,a\right]$ and $\left[1,a+b,3\right]$, and the direction $⟨0,0,1⟩$, with the command $\mathrm{Plane}\left(\left[2,3,a\right],\left[1,a+b,3\right],⟨0,0,1⟩\right)$.
 These parameters must be different from the coordinate variables (set by the variables option explained below; by default $\left[x,y,z\right]$).
 • The following is a list of commands available to Plane

 • In addition, there is some special behavior for the eval command when applied to a Plane. If the plane contains parameters, as in the example $\mathrm{Plane}\left(\left[2,3,a\right],\left[1,a+b,3\right],⟨0,0,1⟩\right)$ above, then one can use eval to substitute values for those parameters. For example, we might evaluate the plane given above at $a=2$ to obtain the line through the points $\left[2,3,2\right]$ and $\left[1,b+2,3\right]$ and containing the direction $⟨0,0,1⟩$.
 Additionally, one can rename the coordinate variables using eval: if we evaluate the plane given above at $\left[x=u,z=v\right]$, then we obtain an identical plane but with the coordinate variables $u$, $y$, and $v$.

Examples

 > $\mathrm{with}\left(\mathrm{Student}\left[\mathrm{MultivariateCalculus}\right]\right):$
 > $\mathrm{p1}≔\mathrm{Plane}\left(3x-y+z+3\right):$$\mathrm{GetRepresentation}\left(\mathrm{p1}\right)$
 ${3}{}{x}{-}{y}{+}{z}{=}{-3}$ (1)
 > $\mathrm{p2}≔\mathrm{Plane}\left(u+v-3w=1,'\mathrm{variables}'=\left[u,v,w\right]\right):$$\mathrm{GetRepresentation}\left(\mathrm{p2}\right)$
 ${u}{+}{v}{-}{3}{}{w}{=}{1}$ (2)
 > $\mathrm{p3}≔\mathrm{Plane}\left(\left[1,2,3\right],\left[1,0,-1\right],\left[0,1,2\right]\right):$$\mathrm{GetRepresentation}\left(\mathrm{p3}\right)$
 ${-}{x}{+}{2}{}{y}{-}{z}{=}{0}$ (3)
 > $\mathrm{p4}≔\mathrm{Plane}\left(\left[1,2,3\right],⟨2,0,-1⟩,'\mathrm{id}=A'\right):$$\mathrm{GetRepresentation}\left(\mathrm{p4}\right)$
 ${2}{}{x}{-}{z}{=}{-1}$ (4)

Note how planes $\mathrm{p1}$ through $\mathrm{p3}$ are displayed using their automatically assigned identification, but $\mathrm{p4}$ has the explicitly specified identification A:

 > $\mathrm{p1},\mathrm{p2},\mathrm{p3},\mathrm{p4}$
 ${\mathrm{<< Plane 1 >>}}{,}{\mathrm{<< Plane 2 >>}}{,}{\mathrm{<< Plane 3 >>}}{,}{\mathrm{<< Plane A >>}}$ (5)
 > $\mathrm{p5}≔\mathrm{Plane}\left(\left[1,2,3\right],\left[1,0,-1\right],⟨1,0,2⟩\right):$$\mathrm{GetRepresentation}\left(\mathrm{p5}\right)$
 ${2}{}{x}{+}{2}{}{y}{-}{z}{=}{3}$ (6)
 > $\mathrm{p6}≔\mathrm{Plane}\left(\left[1,2,3\right],⟨0,a,2⟩,⟨a,0,b⟩\right):$$\mathrm{GetRepresentation}\left(\mathrm{p6}\right)$
 ${-}{a}{}{z}{+}{b}{}{x}{+}{2}{}{y}{=}{-}{3}{}{a}{+}{b}{+}{4}$ (7)
 > $\mathrm{l1}≔\mathrm{Line}\left(\left[1,2,3\right],\left[0,1,3\right]\right):$$\mathrm{GetRepresentation}\left(\mathrm{l1}\right)$
 ${t}{·}\left[\begin{array}{c}{-1}\\ {-1}\\ {0}\end{array}\right]{+}\left[\begin{array}{c}{1}\\ {2}\\ {3}\end{array}\right]$ (8)
 > $\mathrm{p7}≔\mathrm{Plane}\left(\left[2,0,1\right],\mathrm{l1}\right):$$\mathrm{GetRepresentation}\left(\mathrm{p7}\right)$
 ${2}{}{x}{-}{2}{}{y}{+}{3}{}{z}{=}{7}$ (9)
 > $\mathrm{l2}≔\mathrm{Line}\left(\left[0,0,0\right],\left[0,1,2\right]\right):$$\mathrm{GetRepresentation}\left(\mathrm{l2}\right)$
 ${t}{·}\left[\begin{array}{c}{0}\\ {1}\\ {2}\end{array}\right]{+}\left[\begin{array}{c}{0}\\ {0}\\ {0}\end{array}\right]$ (10)
 > $\mathrm{l3}≔\mathrm{Line}\left(\left[0,0,1\right],\left[0,1,3\right]\right):$$\mathrm{GetRepresentation}\left(\mathrm{l3}\right)$
 ${t}{·}\left[\begin{array}{c}{0}\\ {1}\\ {2}\end{array}\right]{+}\left[\begin{array}{c}{0}\\ {0}\\ {1}\end{array}\right]$ (11)
 > $\mathrm{p8}≔\mathrm{Plane}\left(\mathrm{l1},\mathrm{l2}\right):$$\mathrm{GetRepresentation}\left(\mathrm{p8}\right)$
 ${-}{2}{}{x}{+}{2}{}{y}{-}{z}{=}{-1}$ (12)
 > $\mathrm{p9}≔\mathrm{Plane}\left(\mathrm{l2},\mathrm{l1}\right):$$\mathrm{GetRepresentation}\left(\mathrm{p9}\right)$
 ${2}{}{x}{-}{2}{}{y}{+}{z}{=}{0}$ (13)
 > $\mathrm{p10}≔\mathrm{Plane}\left(\mathrm{l2},\mathrm{l3}\right):$$\mathrm{GetRepresentation}\left(\mathrm{p10}\right)$
 ${-}{x}{=}{0}$ (14)
 > $\mathrm{p11}≔\mathrm{Plane}\left(\left[1,2,3\right],\mathrm{p10}\right):$$\mathrm{GetRepresentation}\left(\mathrm{p11}\right)$
 ${-}{x}{=}{-1}$ (15)
 > $\mathrm{p12}≔\mathrm{Plane}\left(\mathrm{p11},'\mathrm{variables}'=\left[u,v,w\right]\right):$$\mathrm{GetRepresentation}\left(\mathrm{p12}\right)$
 ${-}{u}{=}{1}$ (16)

eval can be used to make substitutions for the parameters in the plane object.

 > $\mathrm{p13}≔\mathrm{eval}\left(\mathrm{p6},a=1\right):$$\mathrm{GetRepresentation}\left(\mathrm{p13}\right)$
 ${b}{}{x}{+}{2}{}{y}{-}{z}{=}{b}{+}{1}$ (17)
 > $\mathrm{p14}≔\mathrm{eval}\left(\mathrm{p6},\left[a=1,b=3\right]\right):$$\mathrm{GetRepresentation}\left(\mathrm{p14}\right)$
 ${3}{}{x}{+}{2}{}{y}{-}{z}{=}{4}$ (18)

Query the properties of individual planes.

 > $\mathrm{GetPoint}\left(\mathrm{p3}\right)$
 $\left[{1}{,}{2}{,}{3}\right]$ (19)
 > $\mathrm{GetNormal}\left(\mathrm{p4}\right)$
 $\left[\begin{array}{c}{2}\\ {0}\\ {-1}\end{array}\right]$ (20)

AreParallel, Intersects, Distance, and GetIntersection find the relationships between two planes.

 > $\mathrm{AreParallel}\left(\mathrm{p8},\mathrm{p9}\right)$
 ${\mathrm{true}}$ (21)
 > $\mathrm{Intersects}\left(\mathrm{p8},\mathrm{p9}\right)$
 ${\mathrm{false}}$ (22)
 > $\mathrm{Intersects}\left(\mathrm{p8},\mathrm{p10}\right)$
 ${\mathrm{true}}$ (23)

The intersection between $\mathrm{p8}$ and $\mathrm{p10}$ is the line $\mathrm{l3}$.

 > $\mathrm{Equal}\left(\mathrm{l3},\mathrm{GetIntersection}\left(\mathrm{p8},\mathrm{p10}\right)\right)$
 ${\mathrm{true}}$ (24)
 > $\mathrm{Distance}\left(\mathrm{p8},\mathrm{p10}\right)$
 ${0}$ (25)
 > $\mathrm{Distance}\left(\mathrm{p8},\mathrm{p9}\right)$
 $\frac{{1}}{{3}}$ (26)

Compatibility

 • The Student[MultivariateCalculus][Plane] command was introduced in Maple 17.