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Math Help - Graphing a vector valued function

  1. #1
    No one in Particular VonNemo19's Avatar
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    Graphing a vector valued function

    Hi.

    I am asked to graph \vec{r}(t)=2t\vec{i}+\sin(4t)\vec{j}+\cos(4t)\vec{  k}. How do I go about this?
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  2. #2
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    Re: Graphing a vector valued function

    I'll assume that \vec{i}, \vec{j} and \vec{k} point along the x-, y- and z-axes, respectively. You see that along the path, y^2+z^2=\sin^2 4t+\cos^2 4t=1, so the graph lies on the cylinder with radius 1 going along the x-axis. The x-coordinate grows linearly. If x(t) were constant, then y(t), z(t) would follow a circle. Therefore, this is a spiral. Further, (y(t), z(t)) make 2 revolutions when t ranges from 0 to \pi, so when x(t) = 2t, there will be two revolutions when x ranges from 0 to 2\pi.

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  3. #3
    Senior Member DeMath's Avatar
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    Re: Graphing a vector valued function

    This is a spiral with the unit radius and the period \frac{\pi}{2}, whose center is the x-axis.
    See this picture (red: t=0\ldots\frac{\pi}{2}, blue: t=\frac{\pi}{2}\ldots\pi)



    For maple

    with(plots):
    A1 := spacecurve([2*t, sin(4*t), cos(4*t)], t=0..(1/2)*Pi, color=red, thickness=3, numpoints=1000):
    A2 := spacecurve([2*t, sin(4*t), cos(4*t)], t=(1/2)*Pi..Pi, color=blue, thickness=3, numpoints=1000):
    display(A1, A2, axes=normal);
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