The PositionVector function constructs a position Vector, 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, position Vectors, rooted Vectors, free Vectors, and vector fields, see VectorCalculus,Details.

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. This has no mathematical meaning in non-cartesian coordinates, so the c parameter only changes the way the components are interpreted. Note that the Student[VectorCalculus] package only supports the cartesian, polar, spherical and cylindrical coordinate systems.

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 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, and 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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$\mathrm{S2}≔\mathrm{PositionVector}\left(\left[1\,p\,q\right]\,\mathrm{spherical}\left[r,\mathrm{\phi },\mathrm{\theta }\right]\right)$

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

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