Anderson Video - Unit Vectors

Professor Anderson
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>> Hello class. Professor Anderson here. Let's talk about unit vectors. This is a way to describe vectors in cartesian coordinate space, and a unit vector is a dimensionless vector. Okay, there's no units associated with it. And, it has a magnitude [writing] equal to one. Okay, and that's why it's called a unit vectors. One unit, alright. So, the only thing that unit vectors do is, they specify direction. [ Writing ] Okay, they specify the direction that you're moving. Now, in our cartesian coordinate system we said we have x, we have y, we have z, and so we'd like to identify unit vectors that correspond to the x-axis, the y-axis, and the z-axis. And, the ones that we choose are i, j, and k. [ Silence ] We write them as i hat, j hat, and k hat, and the hat; this thing is called a hat, it means it is a unit vector [writing], so it is a vector, but it only has magnitude one. Alright, so anytime you're along the x-axis you're going to use an i hat. Anytime you're along the y-axis you're going to use a j hat. So, let's draw a coordinate system. [ Writing ] And, let's try the following. I'm going to give you a point right here. Let's say that that is point five, four, okay. It is five out in the x direction. One, two, three, four, five. It is four up in the y direction. So, let me ask you the following question. How do I write that vector, and let's call it vector A, how do I write that in terms of unit vectors? And, if somebody knows just raise your hand and shout it out. [ Background Comments ] How do I write the vector A in terms of unit vectors? Any thoughts [silence]? I'll do the hard part for you, okay? [ Writing ] Two dimensions. There's something that's going to go in front of the i hat. There's something else that's going to go in front of the j hat. What do you think? >> (student speaking) A sub x, A sub y. >> A sub x, and A sub y. That sounds fantastic. And, in fact, for the problem that we just identified can we say exactly what those A sub x's and those A sub y's are? So, what would you write? What's the next line I should put? >> (student speaking) A sub x, i would go along the x-axis? >> Good. So, we need to figure out how long this is. How long is that? >> (student speaking) Oh, five. >> Five. Five i hat. If that was five, what does the other one have to be? >> (student speaking) Four. >> Four [writing], okay. That's what this vector is. That's how you write it. Five i hat, plus four j hat, and what it means is let's move five units in the x direction, that's what the i hat means, and let's move four units in the y-axis direction which is the j hat, okay. That's what those two things mean, and that is this vector right here. It starts at the origin, and it goes up to that point five comma four. Okay, everybody okay with that so far? Alright, good.
>> Hello class. Professor Anderson here. Let's talk about unit vectors. This is a way to describe vectors in cartesian coordinate space, and a unit vector is a dimensionless vector. Okay, there's no units associated with it. And, it has a magnitude [writing] equal to one. Okay, and that's why it's called a unit vectors. One unit, alright. So, the only thing that unit vectors do is, they specify direction. [ Writing ] Okay, they specify the direction that you're moving. Now, in our cartesian coordinate system we said we have x, we have y, we have z, and so we'd like to identify unit vectors that correspond to the x-axis, the y-axis, and the z-axis. And, the ones that we choose are i, j, and k. [ Silence ] We write them as i hat, j hat, and k hat, and the hat; this thing is called a hat, it means it is a unit vector [writing], so it is a vector, but it only has magnitude one. Alright, so anytime you're along the x-axis you're going to use an i hat. Anytime you're along the y-axis you're going to use a j hat. So, let's draw a coordinate system. [ Writing ] And, let's try the following. I'm going to give you a point right here. Let's say that that is point five, four, okay. It is five out in the x direction. One, two, three, four, five. It is four up in the y direction. So, let me ask you the following question. How do I write that vector, and let's call it vector A, how do I write that in terms of unit vectors? And, if somebody knows just raise your hand and shout it out. [ Background Comments ] How do I write the vector A in terms of unit vectors? Any thoughts [silence]? I'll do the hard part for you, okay? [ Writing ] Two dimensions. There's something that's going to go in front of the i hat. There's something else that's going to go in front of the j hat. What do you think? >> (student speaking) A sub x, A sub y. >> A sub x, and A sub y. That sounds fantastic. And, in fact, for the problem that we just identified can we say exactly what those A sub x's and those A sub y's are? So, what would you write? What's the next line I should put? >> (student speaking) A sub x, i would go along the x-axis? >> Good. So, we need to figure out how long this is. How long is that? >> (student speaking) Oh, five. >> Five. Five i hat. If that was five, what does the other one have to be? >> (student speaking) Four. >> Four [writing], okay. That's what this vector is. That's how you write it. Five i hat, plus four j hat, and what it means is let's move five units in the x direction, that's what the i hat means, and let's move four units in the y-axis direction which is the j hat, okay. That's what those two things mean, and that is this vector right here. It starts at the origin, and it goes up to that point five comma four. Okay, everybody okay with that so far? Alright, good.