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VectorCalculus

 RootedVector
 create 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); specify root point of the vector vspace - root=module(Vector,GetRootPoint), specify VectorSpace where the vector lies comps - list(algebraic) or Vector(algebraic); components specifying the coefficients of the basis vectors c - (optional) symbol or symbol[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. The rooted Vector is one of the principal data structures of the Vector Calculus package.
 • 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 argument c can be of the form coords=coord_name, coordinates=coord_name,  or simply coord_name, where coord_name is the name of a valid coordinate system.
 • The components comps should be specified as a list. Alternatively, for convenience, it can be specified as a free Vector in Cartesian coordinates or a position Vector. 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.
 • Rooted Vectors always display as column vectors, independent of any formatting specified by a call to the BasisFormat command.
 • For details on the differences between rooted Vectors, free Vectors, position Vectors and vector fields, see VectorCalculus,Details.

Examples

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

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)
 > $\mathrm{v2}≔\mathrm{RootedVector}\left(\mathrm{point}=\left[1,\mathrm{\pi }\right],\left[1,1\right],\mathrm{coords}=\mathrm{polar}\left[r,t\right]\right)$
 ${\mathrm{v2}}{≔}\left[\begin{array}{c}{1}\\ {1}\end{array}\right]$ (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)
 > $\mathrm{vs}≔\mathrm{VectorSpace}\left(\left[2,3\right],\mathrm{parabolic}\left[u,v\right]\right):$
 > $\mathrm{v4}≔\mathrm{RootedVector}\left(\mathrm{root}=\mathrm{vs},\left[1,2\right]\right)$
 ${\mathrm{v4}}{≔}\left[\begin{array}{c}{1}\\ {2}\end{array}\right]$ (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)
 > $\mathrm{GetRootPoint}\left(\mathrm{v4}\right)$
 $\left({2}\right){{e}}_{{u}}{+}\left({3}\right){{e}}_{{v}}$ (13)
 > $\mathrm{GetRootPoint}\left(\mathrm{v5}\right)$
 $\left({2}\right){{e}}_{{u}}{+}\left({3}\right){{e}}_{{v}}$ (14)