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The PositionVector procedure constructs a position Vector, one of the principal data structures of the Vector Calculus package.

The call PositionVector(comps, c) returns a position Vector in a Cartesian enveloping space with components interpreted using the corresponding transformations from c coordinates to Cartesian coordinates.

If no coordinate system argument is present, the components of the position Vector are interpreted in the current coordinate system (see SetCoordinates ).

The position Vector is a Cartesian Vector rooted at the origin, and has no mathematical meaning in non-Cartesian coordinates.

The c parameter specifies the coordinate system in which the components are interpreted; they will be transformed into Cartesian coordinates.

For more information about coordinate systems supported by VectorCalculus, see coords .

If comps has indeterminates representing parameters, the position Vector serves to represent a curve or a surface.

To differentiate a curve or a surface specified via a position Vector, use diff .

To evaluate a curve or a surface given by a position Vector, use eval .

To evaluate a vector field along a curve or a surface given by a position Vector, use evalVF .

A curve or surface given by a position Vector can be plotted using PlotPositionVector .

The position Vector is displayed in column notation in the same manner as rooted Vectors are, as a position Vector can be interpreted as a Vector that is (always) rooted at the Cartesian origin.

A position Vector cannot be mapped to a basis different than Cartesian coordinates. In order to see how the same position Vector would be described in other coordinate systems, use GetPVDescription .

Standard binary operations between position Vectors like +/-,*, Dot Product , Cross Product are defined.

Binary operations between position Vectors and vector fields, free Vectors or rooted Vectors are not defined; however, a position Vector can be converted to a free Vector in Cartesian coordinates via ConvertVector .

For details on the differences between position Vectors, rooted Vectors and free Vectors, see VectorCalculus,Details .

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

$\mathrm{pv1}≔\mathrm{PositionVector}\left(\left[1\,2\,3\right]\,\mathrm{cartesian}\left[x,y,z\right]\right)$

$\mathrm{pv1}≔\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]$

$\mathrm{About}\left(\mathrm{pv1}\right)$

$\left[\begin{array}{cc}\mathrm{Type:}& \mathrm{Position\; Vector}\\ \mathrm{Components:}& \left[1\,2\,3\right]\\ \mathrm{Coordinates:}& {\mathrm{cartesian}}_{x,y,z}\\ \mathrm{Root\; Point:}& \left[0\,0\,0\right]\end{array}\right]$

$\mathrm{PositionVector}\left(\left[1\,\frac{\mathrm{\pi }}{2}\right]\,\mathrm{polar}\left[r,t\right]\right)$

$\left[\begin{array}{c}0\\ 1\end{array}\right]$

$\mathrm{PositionVector}\left(\left[1\,3\right]\,\mathrm{parabolic}\left[u,v\right]\right)$

$\left[\begin{array}{c}\mathrm{-4}\\ 3\end{array}\right]$

$\mathrm{R1}≔\mathrm{PositionVector}\left(\left[p\,p^2\right]\,\mathrm{cartesian}\left[x,y\right]\right)$

$\mathrm{R1}≔\left[\begin{array}{c}p\\ p^2\end{array}\right]$

$\mathrm{PlotPositionVector}\left(\mathrm{R1}\,p=1..2\right)$

$\mathrm{R2}≔\mathrm{PositionVector}\left(\left[v\,v\right]\,\mathrm{polar}\left[r,\mathrm{\theta }\right]\right)$

$\mathrm{R2}≔\left[\begin{array}{c}v\cos \left(v\right)\\ v\sin \left(v\right)\end{array}\right]$

$\mathrm{PlotPositionVector}\left(\mathrm{R2}\,v=0..3\mathrm{\pi }\right)$

$\mathrm{R3}≔\mathrm{PositionVector}\left(\left[1\,\frac{\mathrm{\pi }}{2}+\arctan \left(\frac{1}{2}t\right)\,t\right]\,\mathrm{spherical}\right)$

$\mathrm{R3}≔\left[\begin{array}{c}\frac{2\cos \left(t\right)}{\sqrt{t^2+4}}\\ \frac{2\sin \left(t\right)}{\sqrt{t^2+4}}\\ -\frac{t}{\sqrt{t^2+4}}\end{array}\right]$

$\mathrm{PlotPositionVector}\left(\mathrm{R3}\,t=0..4\mathrm{\pi }\right)$

$\mathrm{S1}≔\mathrm{PositionVector}\left(\left[t\,\frac{v}{\mathrm{sqrt}\left(1+t^2\right)}\,\frac{vt}{\mathrm{sqrt}\left(1+t^2\right)}\right]\,\mathrm{cartesian}\left[x,y,z\right]\right)$

$\mathrm{S1}≔\left[\begin{array}{c}t\\ \frac{v}{\sqrt{t^2+1}}\\ \frac{vt}{\sqrt{t^2+1}}\end{array}\right]$

$\mathrm{PlotPositionVector}\left(\mathrm{S1}\,t=-3..3\,v=-3..3\right)$

$\mathrm{S2}≔\mathrm{PositionVector}\left(\left[1\,p\,q\right]\,\mathrm{toroidal}\left[r,p,t\right]\right)$

$\mathrm{S2}≔\left[\begin{array}{c}\frac{\sinh \left(p\right)\cos \left(q\right)}{\cosh \left(p\right)-\cos \left(1\right)}\\ \frac{\sinh \left(p\right)\sin \left(q\right)}{\cosh \left(p\right)-\cos \left(1\right)}\\ \frac{\sin \left(1\right)}{\cosh \left(p\right)-\cos \left(1\right)}\end{array}\right]$

$\mathrm{PlotPositionVector}\left(\mathrm{S2}\,p=0..2\mathrm{\pi }\,q=0..2\mathrm{\pi }\right)$