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Student[VectorCalculus]

 RootedVector
 creates a vector rooted at a given point with specified components in a given coordinate system

 Calling Sequence RootedVector(origin,comps,c) RootedVector(vspace, comps)

Parameters

 origin - root=list(algebraic) or root=Vector(algebraic); root point of the vector vspace - root=module(Vector,GetRootPoint), VectorSpace where the vector lies comps - list(algebraic) or Vector(algebraic); components specifying the coefficients of the basis vectors c - (optional) name or name[name, name, ...]; specify the coordinate system, possibly indexed by the coordinate names

Description

 • The call RootedVector(origin,comps,c) returns a vector rooted at point origin with components comps in c coordinates. Note that the Student[VectorCalculus] package only supports the cartesian, polar, spherical and cylindrical coordinate systems.
 • The rooted Vector is one of the four principal Vector data structures of the Student[VectorCalculus] package. Note that the Student[VectorCalculus] and the VectorCalculus packages share the same Vector data structures.
 • For details on the differences between the four principal Vector data structures, namely, rooted Vectors, position Vectors, free Vectors, and vector fields, see VectorCalculus,Details.
 • If no coordinate system argument is present, the current coordinate system is used.
 • The root point origin can be specified as a free or position Vector or as a list of coordinate entries. If it is a free or position Vector, the coordinate system attribute is checked and conversion of the point to the current or specified c coordinate system is done accordingly.
 • The keyword root can also be given as point.
 • The components comps must be specified as a free Vector in Cartesian coordinates, a position Vector or as a list.  The elements of the Vector or list are taken to be the coefficients of the unit basis vectors in the target coordinate system (as specified by the c parameter, if given, or else the current coordinate system).
 • The call RootedVector(vpsace,comps) returns a vector rooted at the root point of the VectorSpace vspace with components comps in the coordinate system of vspace. No extra coordinate system needs to be specified.  The comps argument can be a list, a free Vector in Cartesian coordinates or a position Vector.
 • The returned rooted vector has a VectorSpace attribute that contains a module representation of the vector space rooted at the point origin.
 • RootedVectors are always displayed as column vectors.

Examples

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

Introductory Examples:

 > $\mathrm{v1}≔\mathrm{RootedVector}\left(\mathrm{root}=\left[1,2,3\right],\left[1,1,1\right]\right)$
 ${\mathrm{v1}}{≔}\left[\begin{array}{c}{1}\\ {1}\\ {1}\end{array}\right]$ (1)
 > $\mathrm{About}\left(\mathrm{v1}\right)$
 $\left[\begin{array}{cc}{\mathrm{Type:}}& {\mathrm{Rooted Vector}}\\ {\mathrm{Components:}}& \left[{1}{,}{1}{,}{1}\right]\\ {\mathrm{Coordinates:}}& {\mathrm{cartesian}}\\ {\mathrm{Root Point:}}& \left[{1}{,}{2}{,}{3}\right]\end{array}\right]$ (2)
 > $\mathrm{GetSpace}\left(\mathrm{v1}\right)$
 ${\mathbf{module}}\left({}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{local}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathrm{_origin}}{,}{\mathrm{_coords}}{,}{\mathrm{_coords_dim}}{;}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{export}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathrm{GetCoordinates}}{,}{\mathrm{GetRootPoint}}{,}{\mathrm{Vector}}{,}{\mathrm{eval}}{;}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{end module}}$ (3)
 > $\mathrm{GetRootPoint}\left(\mathrm{v1}\right)$
 $\left({1}\right){{e}}_{{x}}{+}\left({2}\right){{e}}_{{y}}{+}\left({3}\right){{e}}_{{z}}$ (4)
 > $\mathrm{v2}≔\mathrm{RootedVector}\left(\mathrm{point}=\left[1,\mathrm{\pi }\right],\left[1,1\right],\mathrm{polar}\left[r,t\right]\right)$
 ${\mathrm{v2}}{≔}\left[\begin{array}{c}{1}\\ {1}\end{array}\right]$ (5)
 > $\mathrm{About}\left(\mathrm{v2}\right)$
 $\left[\begin{array}{cc}{\mathrm{Type:}}& {\mathrm{Rooted Vector}}\\ {\mathrm{Components:}}& \left[{1}{,}{1}\right]\\ {\mathrm{Coordinates:}}& {{\mathrm{polar}}}_{{r}{,}{t}}\\ {\mathrm{Root Point:}}& \left[{1}{,}{\mathrm{\pi }}\right]\end{array}\right]$ (6)
 > $\mathrm{GetCoordinates}\left(\mathrm{v2}\right)$
 ${{\mathrm{polar}}}_{{r}{,}{t}}$ (7)
 > $\mathrm{rp}≔\mathrm{Vector}\left(\left[1,1\right],\mathrm{coords}=\mathrm{cartesian}\right)$
 ${\mathrm{rp}}{≔}\left({1}\right){{e}}_{{x}}{+}\left({1}\right){{e}}_{{y}}$ (8)
 > $\mathrm{v3}≔\mathrm{RootedVector}\left(\mathrm{root}=\mathrm{rp},\left[1,0\right],\mathrm{polar}\left[r,t\right]\right)$
 ${\mathrm{v3}}{≔}\left[\begin{array}{c}{1}\\ {0}\end{array}\right]$ (9)
 > $\mathrm{GetRootPoint}\left(\mathrm{v3}\right)$
 $\left(\sqrt{{2}}\right){{e}}_{{r}}{+}\left(\frac{{\mathrm{\pi }}}{{4}}\right){{e}}_{{t}}$ (10)
 > $\mathrm{vs}≔\mathrm{VectorSpace}\left(\left[2,3\right],\mathrm{polar}\left[r,t\right]\right):$
 > $\mathrm{v4}≔\mathrm{RootedVector}\left(\mathrm{root}=\mathrm{vs},\left[1,2\right]\right)$
 ${\mathrm{v4}}{≔}\left[\begin{array}{c}{1}\\ {2}\end{array}\right]$ (11)
 > $\mathrm{v5}≔\mathrm{RootedVector}\left(\mathrm{root}=\mathrm{vs},\left[3,4\right]\right)$
 ${\mathrm{v5}}{≔}\left[\begin{array}{c}{3}\\ {4}\end{array}\right]$ (12)
 > $\mathrm{GetRootPoint}\left(\mathrm{v4}\right)$
 $\left({2}\right){{e}}_{{r}}{+}\left({3}\right){{e}}_{{t}}$ (13)
 > $\mathrm{GetRootPoint}\left(\mathrm{v5}\right)$
 $\left({2}\right){{e}}_{{r}}{+}\left({3}\right){{e}}_{{t}}$ (14)