The ProjectionPlot(V, S) command plots the projection of the 2- or 3-D Vector V onto the subspace defined by the parameter S.

If the subspace S is defined by a Vector or set or list of Vectors, the subspace is taken to be the span of these Vectors.  If S is an equation, the subspace is taken to be all vectors satisfied by that equation.  A line or plane defined by an equation must pass through the origin.  If S is given as an algebraic expression, it is treated as the equation $S=0$.

If an equation is used to define the line or plane (the subspace), a list of the variable names of the axes must be given. The list is not required when all axis variable names appear in the expression, in which case, they are sorted lexicographically if no order is given.  For example, given the equation $3x+z=4y$, the order chosen is $\left[x\,y\,z\right]$.

The opts argument can contain any of the Student plot options or any of the following equations that set plot options.

showbasis = true or false

Specifies whether the Vectors specified in the subspace parameter S are plotted.  This option is ignored if S is given as an equation or algebraic expression.  [Default: false]

basisoptions = list

Provides options (such as color) used to plot the Vectors given in the parameter S.  This option is ignored if S is given as an equation or algebraic expression.  See plots[arrow] for further information.

showcomplement = true or false

Specifies whether the projection of the input Vector V onto the orthogonal complement of S is plotted.  [Default: true]

complementoptions = list

Provides options (such as color) used to plot the projection of V onto the orthogonal complement of S.  See plots[arrow] for further information.

showlines = true or false

Specifies whether lines from the head of V to the heads of the projections of V onto the subspace S and its orthogonal complement are plotted.  This option is subordinate to each of the showcomplement and showprojection options.  [Default: true]

lineoptions = list

Provides options (such as color, linestyle) used to plot the lines drawn from the head of V to the heads of the projections of V onto the subspace S and its orthogonal complement.  See plot/options for further information.

showprojection = true or false

Specifies whether the projection of the input Vector V onto the subspace S is plotted.  [Default: true]

projectionoptions = list

Provides options (such as color) used to plot the projection of V onto the subspace S.  See plots[arrow] for further information.

showsubspace = true or false

Specifies whether the subspace defined by the parameter S is plotted.  [Default: true]

subspaceoptions = list

Provides options (such as color, style) used to plot the subspace S. See plot/options and plot3d/options for further information.

showvector = true or false

Specifies whether the Vector being projected, V, is plotted. [Default: true]

vectoroptions = list

Provides options (such as color) used to plot the input Vector V. See plots[arrow] for further information.

caption = anything

A caption for the plot.

The default caption is constructed from the parameters and the command options. caption = "" disables the default caption. For more information about specifying a caption, see plot/typesetting.

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{LinearAlgebra}\right]\right)\:$

$\mathrm{infolevel}\left[\mathrm{Student}\left[\mathrm{LinearAlgebra}\right]\right]≔1\:$

$\mathrm{ProjectionPlot}\left(⟨-2\,3⟩\,⟨7\,-3⟩\right)$

Vector:     <-2, 3>
Projection: <-2.776, 1.190>
Orthogonal complement: <.7759, 1.810>
Norm of orthogonal complement: 1.970

$\mathrm{ProjectionPlot}\left(⟨-3\&comma;5\&comma;-3⟩\&comma;x+3y=0\&comma;\left[x\&comma;y\&comma;z\right]\&comma;\mathrm{showcomplement}=\mathrm{false}\right)$

Vector:     <-3, 5, -3>
Projection: <-4.200, 1.400, -3.>
Orthogonal complement: <1.200, 3.600, 0.>
Norm of orthogonal complement: 3.795
Equation of the plane: x+3*y = 0

$\mathrm{ProjectionPlot}\left(⟨3\&comma;2⟩\&comma;⟨5\&comma;-3⟩\right)$

Vector:     <3, 2>
Projection: <1.324, -.7941>
Orthogonal complement: <1.676, 2.794>
Norm of orthogonal complement: 3.258

$\mathrm{ProjectionPlot}\left(⟨1\&comma;2\&comma;3⟩\&comma;\left\{⟨1\&comma;1\&comma;1⟩\&comma;⟨1\&comma;-1\&comma;-1⟩\right\}\&comma;\mathrm{showbasis}\&comma;\mathrm{lineoptions}=\left[\mathrm{color}=\mathrm{magenta}\right]\&comma;\mathrm{orientation}=\left[-25\&comma;75\right]\right)$

Vector:     <1, 2, 3>
Projection: <1.000, 2.500, 2.500>
Orthogonal complement: <.7949e-13, -.5000, .5000>
Norm of orthogonal complement: .7071