# Superimposing Manipulate on a Static Plot

I'm new (I started today) to Mathematica so

1. Please forgive me if my code is inefficient/godawful and
2. I realize my question has already sort of been asked (combining manipulate with "static" plot), but I can not make heads or tails of the answer provided to that question, as I am obviously not really familiar with Mathematica whatsoever, so here I go.

I am basically trying to use Superimpose a unit tanjent vector (tanjent vector being manipulated) onto a static 3D parametric plot, and I am not sure how to do so. When I attempted to use Show with the Manipulate graphic as one argument and the ParametricPlot3D as another argument it does not display them together and says "Could not display the two graphics together".

My Parametric equation is:

curve[u_] := {Cos[u]*u^.5, Sin[u]*u^.5, u*Sin[u]^2}


And the Parametric plot I attempt to Show it as with my manipulate graphic is:

ParametricPlot3D[curve[u], {u, 0, 20}]


My Manipulate graphic code is:

Manipulate[
Graphics3D[{Green,
Arrow[{{Cos[a]*a^.5, Sin[a]*a^.5,
a*Sin[a]^2}, {Cos[a]*a^.5 + xPrime[a]/lengthOf[a],
Sin[a]*a^.5 + yPrime[a]/lengthOf[a],
a*Sin[a]^2 + zPrime[a]/lengthOf[a]}}]}, Axes -> True,
PlotRange -> {{-4, 4}, {-4, 4}, {0, 20}}], {a, 0, 20}]


And the background code for Manipulate is:

xPrime[u_] := -(u^.5)*Sin[u] + .5*(u^-.5)*Cos[u]
yPrime[u_] := (u^.5)*Cos[u] + .5*(u^-.5)*Sin[u]
zPrime[u_] := 2*u*Sin[u]*Cos[u] + Sin[u]^2
lengthOf[u_] := ((xPrime[u])^2 + (yPrime[u])^2 + (zPrime[u])^2)^.5


It would be very helpful if someone could explain how to superimpose the two graphics, as I'll probably be doing more of this. Thank you in advance.

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A problem is the fact that $a=0$ is not an acceptable value for $a$, as it leads to division by zero; for your purposes, any value very close to zero, but not zero, will do. Secondarily, it seems easiest to nest your Show expression inside the Manipulate.

Manipulate[
Show[
ParametricPlot3D[curve[u], {u, 0, 20}],
Graphics3D[{Green,
Arrow[{{Cos[a]*a^.5, Sin[a]*a^.5,
a*Sin[a]^2}, {Cos[a]*a^.5 + xPrime[a]/lengthOf[a],
Sin[a]*a^.5 + yPrime[a]/lengthOf[a],
a*Sin[a]^2 + zPrime[a]/lengthOf[a]}}]
}, Axes -> True, PlotRange -> {{-4, 4}, {-4, 4}, {0, 20}}
]
],
{a, 0.01, 20}
] • That seems to solve my problem perfectly, thank you very much MarcoB, and thank you for the welcome Louis. – TMathematica Nov 26 '16 at 17:48