# How do you manipulate this in 2D

Use Manipulate to create an interactive plot that shows the parametrically-defined curve described by {x[t], y[t]} = {t Sin[t], t^2/15} for {t,0,10} along with a point of tangency, tangent vector, and tangent line to the curve for any choice of t between 0 and 10.

Clear[t, x, y, P, velvector, vel, scalefactor];
scalefactor = 0.5;
P[t_] = {t Sin[t], t^2/15};
curveplot = ParametricPlot[P[t], {t,0,10}, PlotStyle -> Thickness[0.01],
AxesLabel -> {"x", "y","z"}];
vel[t_] = D[P[t], t];
velvector[t_] := Vector[vel[t], Tail -> P[t], VectorColor -> Red, ScaleFactor
-> scalefactor];
Manipulate[Show[curveplot, velvector[t],{t,0,10}, PlotRange -> All]]


Something not right here, it failed.

• ...and do you have any code showing your attempt at solving this? – J. M.'s technical difficulties Jun 28 '16 at 2:01
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• Most users on this site will not be too interested in just doing a problem for someone, especially if it's homework (many of us are educators, so we don't look kindly on people asking us to do their homework). What exactly about this problem is giving you trouble? Is it how to use Manipulate (see the documentation)? Is it how to plot a point, line, and vector on top of a graph (construct the objects using math and use Graphics to make the pictures, then use Show: see the docs)? Is it how to make a parametric plot to begin with (use the docs)? Be specific, and show your work. – march Jun 28 '16 at 2:37
• ​Clear[t, x, y, P, velvector, vel, scalefactor]; scalefactor = 0.5; P[t_] = {t Sin[t], t^2/15}; curveplot = ParametricPlot[P[t], {t,0,10}, PlotStyle -> Thickness[0.01], AxesLabel -> {"x", "y","z"}] vel[t_] = D[P[t], t]; velvector[t_] := Vector[vel[t], Tail -> P[t], VectorColor -> Red, ScaleFactor -> scalefactor]; From here I have my vector and my curve. How do i plot it in parametrization of using manipulate with the constraint of t? Manipulate[Show[curveplot, velvector[t],{t,0,3}, PlotRange -> All]] – user41325 Jun 28 '16 at 3:02
• Please edit your question to include the code in your comment. – J. M.'s technical difficulties Jun 28 '16 at 3:26

I guess what you are looking for is something like this

scalefactor = 0.5;

P[t_] = {t Sin[t], t^2/15};

curveplot[t0_] := ParametricPlot[P[t], {t, 0, t0},
PlotStyle -> Thickness[0.01], AxesLabel -> {"x", "y"}]

vel[t_] = D[P[t], t];
velvector[t0_] := Graphics[{Red, Arrow[{P[t0], P[t0] + scalefactor vel[t0]}]}]

Manipulate[Show[curveplot[t], velvector[t], PlotRange -> {{0, 2}, {0, 1}},
AspectRatio -> 1], {t, 1, 10}]