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Sketch the curve with the given vector equation. Indicate with an arrow the direction in which $ t $ increases.

$ r(t) = 2 \cos t i + 2 \sin t j + k $

circle with radius 2 centered at $(0,0,1)$ on the plane $z=1$

Vector Functions

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in this problem. We have a equation which we are trying to grab the equation is R. T. Is equal to two coast T. I had plus to sign T. J. Had plus K. Hat. So to grab this, what we're going to do is we're going to break it out into its components and what those are individually starting with to coast T. I had so we know that this is just a coast function magnified by two. So I'm gonna go ahead and grab this here between zero T and two pi T. And to Christie we should now start at two versus zero -20 back to positive too. So this year is how are I had well act we'll go ahead and do the same thing for R. J. Hat which is to sign of T. And just like a regular. Just as you'd imagine the sine function goes from zero to kill zero negative two. Zero. Look something like this and the last component the k hat is actually quite simple. It is simply one plus K had provision one and just be constant. Now to compose this photograph, what we're gonna do is we're going to look at at different points in the cycle as you can tell as you should be able to tell this is a cyclical graph. It started Its position zero T is the same position at T at your property. So after to pry of T. Everything will just repeat. So starting at 0-2000. Um I had is equal to two. Okay is equal to one and a sign is equal to zero. We're actually going to go ahead and plot this point on our mm laugh the curve. So I had to equal to two. Is it for the 1? And so your first point we're gonna go ahead to the second point which is that I over to T. You're I had to call the zero. Jihad is equal to two and okay Hot is still equal to one. So we don't go now I have so we just get you in the jihad Direction one in the case that direction. This will be a second point and they'll just go here as a regular circle. It's a very good circle but it goes in this direction as he is increasing as it goes from this point. At this point we're going to put an arrow the direction that T increases. Then when she is able to play RJ had directions again equal to zero And I had is now equal to -2. But I had people negative too. Okay, it's illegal. The one and Now right before we finish the cycle we go three pi over two. Yeah I had is equal to zero. They had is equal to -2. Think he had is equal for one. Still as always and this will be that point on the graph and then as we said, when it reaches two pi it actually reaches the exact same position. It was at at T equals zero. The graph should look something like this. You're a better artist than me that will look like a circle as the coast sea of society are perfectly cyclical functions. So I hope this was very helpful. Thank you.