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

$$\mathbf{r}(t)=t \mathbf{i}+t \mathbf{j}+t \mathbf{k}, \quad 0 \leq t \leq 2$$

Graph (d)

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Okay, folks. So in this video, we're gonna have this victory equation that we want to match with one of the grafts from eight age. Okay, so let me write down the specter equation for you. We have are as a function of team, which is a vector value to function and is going to give you tea in the X component plus t. Why? Component plus t um, in the Z component. And we're gonna be restricted, um, with with with the following restriction, which is zero less than or equal to see less than or equal to two. Um so so one of the There's a couple of things to notice here. The first thing is, the first thing is the fact that time that we have three linear functions. There's no teach, squared or too cute or whatever. We have three simple looking, nice linear functions. So and and and what's what's the most important with this? Is that that the x component and why component and the Z component of this vector valued function is the same. So we have X equals y equals E. So all of our points that's gonna be that's going to be drawn on. The graph has equal values for the X and Y and Z components. So basically, what I'm saying is it's supposed to be aligned and it's allying with with with some very nice properties. For example, the three components are are the same and it starts off at the origin because when t zero all of the components are zero and so weaken, graph it out, we can go ahead and see if we can grab it up. So this is some X and this is why. And this is Z when 20 0 we are located at the origin. So that's the origin. That's when to you zero. And when tea is too, we have X equals Why you cause the Eagles to So I'm gonna graft that out. We have to here and two here and two right here. Let's see if I can grab it out. Um, there we go. So this is our our function are of tea. It's basically just a line. You know, I'm not grabbing it very well. I can't draw, but I hope you understand. What I'm trying to say here when I'm trying to say is that it's supposed to be a line and and if you look through all of the options from age, you'll see that the only option given to us that matches our description is the option. D and we can do a quick check. Um, for example, Part D says that that the end point of this line segment of its line segment is 2 to 2. And that's exactly where we ended up with with our function or if t because our function when tea is too. Our function has X and Y and Z all equaling two. So this point, what I'm saying here is that this point is to to to and that matches perfectly with with D. And so that's it for this video. Because this is the answer. The answer is deep, all right?

University of California, Berkeley