Plotting Vector and Derivative: A Frustrating Challenge

[dpeagler]

I have been trying to plot a vector and it's derivative for a while now, but I cannot figure out why I can't get them to plot.

Here is what I put for my code...

a[t_] := Sin[t] Cos[2 t]
b[t_] := Sin[t] Sin[2 t]
c[t_] := Cos[t]
V[t_] := {a, b, c}

I tried this...

ParametricPlot[{a[t], b[t], c[t]}, {a'[t], b'[t], c'[t]}, {t, -\[Pi]/2, \[Pi]/2}, {t, \[Pi]/4}]

and this...

ParametricPlot[{V[t]},{V'[\[Pi]/4]},{{t, -\[Pi]/2, \[Pi]/2}

The function is seen as V[t]. I want to graph it from -Pi/2 to Pi/2 and I want to have V'[Pi/4] on the same graph. And then I want to have two different images.

I know that's a lot, but I'm sure it's something really simple that I'm missing. Thanks to anyone who can give any advice.

 

[Bill Simpson]

See if you can just plot the vector first.

a[t_] := Sin[t] Cos[2 t];
b[t_] := Sin[t] Sin[2 t];
c[t_] := Cos[t];
V[t_] := {a[t], b[t], c[t]};
ParametricPlot3D[V[t], {t, -Pi/2, Pi/2}]

If that works then compare what I did with what you did and figure out the differences.

After that see if you can just plot the derivative

ParametricPlot3D[{a'[t], b'[t], c'[t]}, {t, -Pi/2, Pi/2}]

After that perhaps you can start thinking about combining the two

 

[dpeagler]

Thanks for the response. The only thing I am confused about is why you are plotting the derivative over the same range. It is only supposed to be at the point Pi/4

 

[Bill Simpson]

See if you can use the methods described here

http://reference.wolfram.com/mathematica/howto/CombineTwoOrMoreGraphics.html

to combine your parametric plot of V[t] with a point in 3D at V'[Pi/4]

 

[blue_raver22]

I can understand your frustration with trying to plot vectors and their derivatives. It can be a challenging task, but with some careful consideration and understanding of the mathematical principles involved, it can be achieved.

Firstly, it's important to note that a vector, such as the one you have defined as V[t], is a function of time. This means that it has a value at every point in time, and therefore cannot be plotted in the same way as a regular function.

To plot a vector, you need to use a parametric plot, which allows you to plot functions of multiple variables. In your code, you have correctly used the ParametricPlot function, but there are a few errors that need to be corrected.

Firstly, when defining a vector function, you need to specify the components separately, rather than using curly brackets. So your code for V[t] should be:

V[t_] := {a[t], b[t], c[t]}

Next, in the ParametricPlot function, you need to specify the range of values for the parameter t. In your case, you want to plot from -π/2 to π/2, so the code should be:

ParametricPlot[V[t], {t, -π/2, π/2}]

To plot the derivative of V[t], you can use the VectorPlot function, which is specifically designed for plotting vector fields. The code for this would be:

VectorPlot[V'[t], {t, -π/2, π/2}]

If you want to plot both the vector and its derivative on the same graph, you can use the Show function, which allows you to combine multiple plots. The code for this would be:

Show[ParametricPlot[V[t], {t, -π/2, π/2}], VectorPlot[V'[t], {t, -π/2, π/2}]]

I hope this helps to clarify the process of plotting vectors and their derivatives. Remember to carefully consider the mathematical principles involved and to double check your code for any errors. Good luck!