In Exercises 1–4, u and v have the same direction. In each exerci...

Question

In Exercises 1–4, u and v have the same direction. In each exercise:Is u = v? Explain.

Accepted Answer

Hey, everyone in this problem, we have the given vectors A and B and we're asked to find out if A is equal to B where A and B have the same direction. OK. So we're given A and B in this diagram here, OK. A is shown in green and it's a vector that points from the point negative 2 19 to the 0.25 19. And then we have vector B shown in red that points from the 0.3 negative two to the 0.30 negative two. And we have two answer traces here. Option A is that A is equal to B and option B is that A is not equal. So we had two vectors. OK? Let's think about what it means for them to be equal. OK. Recall that for a vector A to be equal to vector B, they need to have the same direction and the same magnet. Now we're told that they have the same direction and, and we can see that from the diagram as well, these are both horizontal vectors pointing to the right, these are parallel and so we know they have the same direction. So the only thing we're left to check is whether they have the same magnitude. OK. So let's start with the magnitude of vector A and the magnitude of vector A recall is gonna be equal to the length of that vector. OK. So the magnitude is equal to the length of our vector A. And in this case, we're given the beginning and end points of our vector A. So it's just gonna be the distance between these two books. We're gonna call the point on the left A one and the point on the right A two. So we have a one is a point negative two comma 19 and A two is a 0.25 comma 19. And the magnitude is gonna be the distance between these two. Now, the distance between two points we're called is gonna be the square root of the distance between the X components. So A two X minus A one X, what the difference between the white people? A two Y minus A one Y squared. And you can write this the other way around. You could have the A one X minus A two X. OK. We're squaring the value. So it doesn't matter which way you do it. But what's important is that it's consistent. OK. Whichever one is the first for the X component has to be the first for the Y component in this book. All right. Yeah, substituting in our values A two X. So we're looking at 0.2 the X value is minus A one X, the X value of A one which is negative two all yes, A two Y, the Y value of A two is 19 minus the Y value of A one which is also 19 all squared. Now, the second part of our expression that's gonna simplify 19 minus 19 is just 00 squared is just zero. OK. So that goes away completely. We only have to worry about this first term we have and I miss a bracket. We need an extra bracket there. So we have 25 minus negative two all squared. Hm. Well, 25 minus negative two, that's gonna be 25 plus two, which is 27. So we get that the magnitude of A is equal to the square root of 27 squared and the square root and the squared are going to cancel out and we are left with the magnitude of A equal to 27. All right. So we have the magnitude of A. Now we need to find the magnitude of B. Remember we're trying to figure out if these vectors are equal, we know they have the same direction. So the question is, do they have the same magnitude? Now, the magnitude of A is 27. And for the magnitude of B, we're gonna do the exact same thing, the magnitude of B is gonna be the distance between those two points and we're gonna call these ones the one and B two. And if we go up B one is going to be the point on the left three, comma negative two and B two is gonna be the point on the right comma negative two. The magnitude of B is gonna be just like a, it's gonna be the square root of V two X minus V one X all squared plus B two, Y minus B one Y all squared. Substituting in our values, we get the square root that's minus three, all squared. What? Negative two minus negative two, all squid. And again, this second term where we have our Y components, negative two minus negative two is gonna give us zero. OK? This makes sense because these are horizontal vectors. OK? So the Y component of each of these points is the same. And so that difference is gonna be zero. OK? So again, we're dealing with just the first term now and when we do that, we're gonna have the square root 30 minus three square. Well, that's gonna be 27 squared again, just like we had with A and so when we take the square root of something squared, we're gonna get just 27 back. So we get that the magnitude of B is equal to 27. So in this problem, we found that the magnitude of A is equal to the magnitude of B. We already knew that the direction of the two vectors is the same which tells us that the vector A is equal to the vector B corresponding with answer choice A thanks everyone for watching. I hope this video helped see you in the next one.

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In Exercises 1–4, u and v have the same direction. In each exercise:Is u = v? Explain.

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