Equivalent vectors

- [Voiceover] I have some example problems here from our equivalent vectors exercise on Khan Academy, so let's go through these and like always, pause the video and see if you can work through them on your own. So this first one says, "Are vectors u and w equivalent?" And so we can see vector u here in blue and vector w right over here. And we have to remember a vector is defined by both having a magnitude and a direction. And so for two vectors to be equivalent they have to have the same magnitude and the same direction. So when I look at these two, they are clearly pointing in different directions. Vector u is pointing to the bottom right, that's the direction it's pointing in, and vector w is pointing to the bottom left, so they definitely aren't equivalent. So they are not equivalent, scratch that out. So they have different directions, different directions. Different directions. And also think about the magnitude. If I just look at the length of the arrows, just eyeballing it, they look pretty close. Let me verify that. So if I'm starting at the initial point for vector u, how much do I move in the x direction? Well in the x direction I go from negative eight to negative three, so I could say my change in x is positive five, my x increases by five as I go from the initial point, from the x coordinate of the initial point to the x coordinate of the terminal point, so that length. The magnitude of just the x component is five. And let's see what happens in the y direction. So in the y direction, I start at y equals negative two, right over here, and then I go down to y equals negative eight. So my change in y, change in y, is equal to negative six, negative six. Also think about this one over here. What's my change in x? What's my change in x? Well I'm starting at x equals eight, and I am going to x equals three. So my change in x is negative five. I'm gonna write that, change in x is equal to, this is my change in x, change in x. And then what's my change in y? Well I start at y is equal to eight and I go down y is equal to two. Do it just like that. So my change in y is equal to negative six. Now based on the changes in x's and y's, I can figure out the magnitude of each of these vectors. The magnitude of u, so I could write it like this, the magnitude of u, sometimes you'll see the notation like this, the magnitude of u, of vector u I should say, is going to be equal to, we're just gonna use the pythagorean theorem here. It's going to be the square root. The length of this hour is just a hypotenuse of the right triangle. So it's going to be the square root of five squared plus negative six squared so it's gonna be the square root of 25 plus 36, which is equal to the square root of, what is that? It's going to be square root of 61. I can make that radical a little bit smaller, square root of 61. Now what about vector w? So the magnitude of vector w is, I could use a double, those double bars. The magnitude of vector w, well that's going to be the square root of negative five squared, which is 25 plus negative six squared, which is going to be 36. Well that's going to be the square root of 61 as well. So they have the same magnitude, just different directions. So let's see, no they have the same magnitude, but different directions. Yup, that's the choice we like. Let's do one more of these. Are vectors u and w equivalent? Alright, so these look pretty equivalent just eyeballing it. It looks like they're pointed in the same direction, they're going from the bottom left to the top right. And it looks like they're the same length, but we can verify them. Once again by looking at their x and y components. So in the x direction, or thinking about how much we change in x and change in y when we go from the initial point to the terminal point. So in the x direction, for vector u, we go from negative seven to negative four, so we increased by three. And what do we do over here? We go from x equals two to x equals five. So our x component once again has a magnitude of three, our change in x is a positive three. And so our change in y, what is that going to be? Well we're going, on this vector, from y equals... Whoops, I'm using the wrong... We're going from y is equal to one to y is equal to six. So the magnitude of the y component, we could say, is five or our change in y is a positive five. And then the change in y over here also is a positive five. We go from negative seven to negative two. It's also a positive five. So notice our change in x is the same, it's a positive three. Our change in y is the same, positive five. So that let's us know that we have the same magnitude and direction. And so these are equivalent, these are equivalent vectors. And, in fact, as we'll see in the future, you can actually denote a vector by its components. We could say that vector u, vecotr u, is the vector five, oh sorry, the vector three comma five, our change in x comma our change in y. And that is exactly the same as vector w. So that is exactly the same as vector w, has the same change in x, same change of y as you go from the initial point to the tip of the arrow, the terminal point.