Graph of r(t) = costi + sintj + costk

[Jared596]

Can someone tell me the general shape of the graph:

r(t) = costi + sintj + costk

I've been told it's an ellipse, but I thought it was a helix...

 

[Mute]

You can tell it can't be a helix because the z-coordinate will return to its original postion when t = 2pi. Looking at just the x and y coordinates, you see that they're just describing the unit circle. If you look at just the y and z coordinates, they also describe the unit circle. The x and z coordinates, on the other hand, just describe a line in the xz plane that goes between x = z = -1 and x = z = 1. So, why not define a new unit vector $\mathsf{h} = \frac{1}{\sqrt{2}}(\mathsf{i} + \mathsf{k})$? This gives you

$$\mathsf{r}(t) = \sqrt{2} \cos t \mathsf{h} + \sin t \mathsf{j}$$

Now do you see why the trajectory is elliptical?

 

[Jared596]

Yes i do. Thanks for your help.