The TangentVector(C, t) calling sequence computes a Vector in the direction of the tangent vector to a curve in R^2 or R^3.

Note: This vector is not normalized by default, so it is a scalar multiple of the unit tangent vector to the curve C. Therefore, by default, the result is generally different from the output of TNBFrame(C, t, tangent=true). If either normalized=true or normalized is given in options, however, the resulting vector will be normalized.

The curve must have exactly two or three components, that is, the curve that the Vector represents is in R^2 or R^3.

If t is not specified, the command tries to determine a suitable variable name by using the components of C.  To do this, it checks all of the indeterminates of type name in the components of C and removes the ones that are determined to be constants.

If the resulting set has a single entry, that entry is the variable name.  If it has more than one entry, an error is raised.

The options arguments primarily control plot options.

output = value, plot, or animation

This option controls the return value of the command.

output = value returns the value of the tangent. Plot options are ignored if output = value.  This is the default value.

output = plot returns a plot of the space curve and the tangent vectors. The number of tangent vectors is specified by the vectors option.

output = animation returns an animation of the space curve and the tangent vectors. The number of tangent vectors is specified by the vectors option.

curveoptions = list

A list of plot options for plotting the space curve. For more information on plotting options, see plot/options. The default value is [].

normalized = truefalse

Either true or false. Specifies whether the tangent vector is to be normalized. The default value is false.

range = realcons..realcons

The range of the independent variable. The default value is 0..5.

vectoroptions = list

A list of plot options for plotting the tangent vectors. For more information on plotting options, see plot/options. The default value is []. Note: Free Vectors and rooted Vectors are plotted using plots[arrow].

vectors = posint

Specifies how many tangent vectors are to be plotted or animated. The default value is 5.

view = [realcons..realcons, realcons..realcons, realcons..realcons] (3-D) or [realcons..realcons, realcons..realcons] (2-D)

Specify the plot view. For more information, see plot3d/options and plot/options.

caption = anything

A caption for the plot.

The default caption is constructed from the parameters and the command options. caption = "" disables the default caption. For more information about specifying a caption, see plot/typesetting.

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

$\mathrm{TangentVector}\left(\mathrm{PositionVector}\left(\left[\cos \left(t\right)\,\sin \left(t\right)\,t\right]\right)\,t\right)$

$\left[\begin{array}{c}-\sin \left(t\right)\\ \cos \left(t\right)\\ 1\end{array}\right]$

$\mathrm{TangentVector}\left(\mathrm{PositionVector}\left(\left[\cos \left(t\right)\,\sin \left(t\right)\,t\right]\right)\,t\,\mathrm{normalized}\right)$

$\left[\begin{array}{c}-\frac{\sqrt{2}\sin \left(t\right)}{2}\\ \frac{\sqrt{2}\cos \left(t\right)}{2}\\ \frac{\sqrt{2}}{2}\end{array}\right]$

$\mathrm{TangentVector}\left(⟨\exp \left(-t\right)\cos \left(t\right)\,\exp \left(-t\right)\sin \left(t\right)\,t⟩\right)$

$\left[\begin{array}{c}-{\ⅇ}^{-t}\left(\cos \left(t\right)+\sin \left(t\right)\right)\\ -{\ⅇ}^{-t}\left(\sin \left(t\right)-\cos \left(t\right)\right)\\ 1\end{array}\right]$

$\mathrm{TangentVector}\left(⟨\cos \left(t\right)\,\sin \left(t\right)\,t⟩\,\mathrm{output}=\mathrm{plot}\,\mathrm{scaling}=\mathrm{constrained}\right)$

$\mathrm{TangentVector}\left(⟨\cos \left(t\right)\,\sin \left(t\right)\,t⟩\,t\,\mathrm{output}=\mathrm{plot}\,\mathrm{range}=0..7\,\mathrm{vectors}=3\,\mathrm{vectoroptions}=\left[\mathrm{color}=\mathrm{red}\right]\,\mathrm{curveoptions}=\left[\mathrm{color}=\mathrm{green}\right]\,\mathrm{scaling}=\mathrm{constrained}\right)$

$\mathrm{TangentVector}\left(⟨\cos \left(t\right)\,\sin \left(t\right)\,t⟩\,t\,\mathrm{output}=\mathrm{animation}\,\mathrm{range}=0..7\,\mathrm{vectors}=3\,\mathrm{vectoroptions}=\left[\mathrm{color}=\mathrm{red}\right]\,\mathrm{curveoptions}=\left[\mathrm{color}=\mathrm{green}\right]\,\mathrm{scaling}=\mathrm{constrained}\right)$

$\mathrm{SetCoordinates}\left(\mathrm{cylindrical}\left[r,t,z\right]\right)$

${\mathrm{cylindrical}}_{r,t,z}$

$\mathrm{simplify}\left(\mathrm{TangentVector}\left(⟨a\,t\,t⟩\right)\right)\mathbf{assuming}a::\left(\mathrm{And}\left(\mathrm{positive}\,\mathrm{constant}\right)\right)$

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

$\mathrm{SetCoordinates}\left(\mathrm{cartesian}\right)\:$

$\mathrm{TangentVector}\left(⟨\cos \left(t\right)\,\sin \left(t\right)\,t⟩\,t\,\mathrm{output}=\mathrm{plot}\,\mathrm{scaling}=\mathrm{constrained}\right)$