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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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