Vectors are quantities that have a magnitude and a direction. In the two-dimensional plane, we can describe them in an equivalent way, by thinking about the changes in x and y from the vector's tail to its head. Created by Sal Khan.
- [Instructor] In other videos, we have talked about how a vector can be completely defined by a magnitude and a direction, you need both. And here we have done that. We have said that the magnitude of vector a is equal to three units, these parallel lines here on both sides, it looks like a double absolute value. That means the magnitude of vector a. And you can also specify that visually by making sure that the length of this vector arrow is three units long. And we also have its direction. We see the direction of vector a is 30 degrees counter-clockwise of due East. Now in this video, we're gonna talk about other ways or another way to specify or to define a vector. And that's by using components. And the way that we're gonna do it is, we're gonna think about the tail of this vector and the head of this vector. And think about as we go from the tail to the head, what is our change in x? And we could see our change in x would be that right over there. We're going from this x value to this x value. And then what is going to be our change in y. And if we're going from down here to up here, our change in y, we can also specify like that. So let me label these. This is my change in x, and then this is my change in y. And if you think about it, if someone told you your change in x and change in y, you could reconstruct this vector right over here by starting here, having that change in x, then having the change in y and then defining where the tip of the vector would be relative to the tail. The notation for this is we would say that vector a is equal to, and we'll have parenthesis, and we'll have our change in x comma, change in y. And so if we wanted to get tangible for this particular vector right over here, we know the length of this vector is three. Its magnitude is three. We know that this is, since this is going due horizontally and then this is going straight up and down. This is a right triangle. And so we can use a little bit of geometry from the past. Don't worry if you need a little bit of a refresher on this, but we could use a little bit of geometry, or a little bit of trigonometry to establish, if we know this angle, if we know the length of this hypotenuse, that this side that's opposite the 30 degree angle is gonna be half the hypotenuse, so it's going to be 3/2. And that the change in x is going to be the square root of three times the 3/2. So it's going to be three, square roots of three over two. And so up here, we would write our x component is three times the square root of three over two. And we would write that the y component is 3/2. Now I know a lot of you might be thinking this looks a lot like coordinates in the coordinate plane, where this would be the x coordinate and this would be the y coordinate. But when you're dealing with vectors, that's not exactly the interpretation. It is the case that if the vector's tail were at the origin right over here, then its head would be at these coordinates on the coordinate plane. But we know that a vector is not defined by its position, by the position of the tail. I could shift this vector around wherever and it would still be the same vector. It can start wherever. So when you use this notation in a vector context, these aren't x coordinates and y coordinates. This is our change in x, and this is our change in y. Let me do one more example to show that we can actually go the other way. So let's say I defined some vector b, and let's say that its x component is square root of two. And let's say that its y component is square root of two. So let's think about what that vector would look like. So it would, if this is its tail, and its x component which is its change in x is square root of two. So it might look something like this. So that would be change in x is equal to square root of two. And then its y component would also be square root of two. So I could write our change in y over here is square root of two. And so the vector would look something like this. It would start here and then it would go over here, and we can use a little bit of geometry to figure out the magnitude and the direction of this vector. You can use the Pythagorean theorem to establish that this squared plus this squared is gonna be equal to that squared. And if you do that, you're going to get this having a length of two, which tells you that the magnitude of vector b is equal to two. And if you wanted to figure out this angle right over here, you could do a little bit of trigonometry or even a little bit of geometry recognizing that this is going to be a right angle right over here, and that this side and that side have the same length. So these are gonna be the same angles which are gonna be 45 degree angles. And so just like that, you could also specify the direction, 45 degrees counter-clockwise of due East. So hopefully you appreciate that these are equivalent ways of representing a vector. You either can have a magnitude and a direction, or you can have your components and you can go back and forth between the two. And we'll get more practice of that in future videos.