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

$$\mathbf{r}(t)=\mathbf{i}+\mathbf{j}+t \mathbf{k}, \quad-1 \leq t \leq 1$$

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Johns Hopkins University

Campbell University

Harvey Mudd College

Boston College

Okay, folks. So in this video, we're gonna take a look at this vector equation, and we want to match it with one of the grafts from eight h. Let me just copy down this defector equation. Here are we have this vector equation that says, um are as a vector value and function gives you, um gives you I Excuse me. I'm the right eye Jane K as X had, Why hadn't and Z had? That's something that I can do something to write X had. Plus, why had plus tee times has he had thes three are just a unit vectors, except why hadn't she had? And our time interval is restricted to from negative one. Um, from negative. One toe one. OK, so now let's take a look at this victor value to function. Let's take a look at its components and see what's going on here. So, first of all, I want you to look at this This x component and why component right here during the coefficient in front of the to get vectors are actually won their constant. They're not changing with time. So what that means is that during your whole time interval your X and Y component They shouldn't change. They should stay one. So the X for all funk for all values of teeth between negative one and one should be one. And why, for all valued for all values of t should be one. And Z, as you can see is a linear function. It grows linearly. What that means is that if we were to graph dysfunction out, this is the X axis. This is the white access and this is the Z axis. And I say that this is the This is the value one here in the value one Here, Um, if you were to graft, X equals wife was one that points going right here. And we want that the value of Z to grow linearly when Z is equal to negative one. Um, the Z component of the function are is gonna be negative one. And so we're gonna start off right here. This is the point that we're going to start off with, and then we're gonna want this function to roll in the earlier we wanted, we wanted to grow straight up. Basically because X and Y are not changing, that's the most important thing. So basically, the function is just gonna look something like this. And this is the component. This is the value of the finals. The value when t is equal to one that Z is gonna be equal to one. Okay, so this is the graph that we want. And if you were to look through the list of options that were given, this matches with the Option E. And so this is the answer. And because that answer, because this matches perfectly with what we have raft here and we're done. Thank you.

University of California, Berkeley