Match the vector equations with the graphs (a)-(h) given here. (GRAPH CANT COPY)

$$\mathbf{r}(t)=(2 \cos t) \mathbf{1}+(2 \sin t) \mathbf{J}, \quad 0 \leq t \leq 2 \pi$$

Graph (g)

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Missouri State University

Oregon State University

University of Michigan - Ann Arbor

Idaho State University

Okay, folks. So in this video, we're gonna take a look at this vector equation, and we want to match this vector equation with one of the options between A and H. So we're we're giving a bunch of graphs. We want to see which one. This victory equation is associated with a case. I'm gonna write down this victory question. We have the function are which is a vector valued function bar amateur ized by the variable t we have to co sign if t excuse me to co sign of t multiplied by I had, which is just the same thing as X had and plus to sign of t multiplied by Jay had, which is the same thing as Whitehead. And our time and time interval is between zero and two pi. Okay, so the first thing I want you to know, this is the fact that we don't have a Z component. And when that happens, what it means is that we're going to restrict our graph to be just the X Y plane. So whatever our final graph is gonna look like, it better not have the Z component that z component that of that function better be zero at all times. So just by that we canoed. Look, we can eliminate a bunch of choices, but yeah, so let's get back to this vector value function here and see what else we can get out of this some expression. Um, one thing I want you to notice is that this is really if you remember your polar coordinates. The X component of the point is usually represented with with polar cornets as our co sign of data. And the white component is represented with our sign. If they don't, I hope you all know this. Um but if you look at this, if you look at what we're given were given to co sign of tea, um, multiplied by X right, That means the X component of our point is going to be to co sign it e. And that looks awfully similar to these equations because they are, they're the same form. And for example, when t zero co sign of tea is one and our X component is too. This is the X axis, this white axis and when t zero sign is signed is zero. So why component is zero. So I want to start off here, and as you can, you can probably tell where I'm going with this. As time grows, our graphics is gonna creates out a circle A circle of radius too. So this is what our graph is going to look like. It's just a circle lying on the X my plane with a radius of two. And it's centered at the origin and with the Z component of zero at all times. So if you look through the options through from H, you're going to see that the only option that matches our description is the option g. So that is going to be our answer. And we're done for this video. Thank you. Bye.

University of California, Berkeley