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Adding Multiples of Vectors

Patrick Ford
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Hey, guys. So up until now we've seen how to simply adventures together like A and B. But sometimes in problems, you're gonna have to add multiples of vectors, like to a plus three B or to a plus 0.5 be something like that. What we're gonna see is that it works exactly like Vector Edition. The only thing that's different is that when you multiply a vector by sticking the number in front of it like a becomes to a, what's happening is because they have a number in front of it. The magnitude of the length is gonna change, but not the direction. Let's check it out. So when we had vectors A and B, basically you would just line them up, tip to tail like this, and your result in vector is the shortest path from the start of the first to the and the last. It's basically is if you had walked in this direction, if you were like calculating the total displacement or something like that and so we just break this up into a triangle and then we count up the boxes to figure out the legs. This is five and five And so the magnitude would just be five squared and five squared. And that's 7.7 What if instead of a plus B, I was given to a plus 0.5 B. Well, one way you can think about this. These multiples air these numbers in front is you can think about this to a here as just being a plus A. So you're still just doing vector addition here. This number here is basically kind of just like condensing all of this information just into a single number. And these numbers are going to change the length of the vectors. So this is a plus a and then you're gonna add it to 05 B, which you can think of is like half of a B. Let's check it out. So if my vector A is three to the rights and one up, then that means that a plus a or to A is just gonna be If I have three of the right and one up like this, and then you just add a tip to tail to another 13 to the right and one up like this, so we're gonna have a and a and basically this whole entire vector here is to a so that if I had added 2.5 b, we do the same thing. So my 0.5 b or sorry, my regular be vector is to to the right and four up. So that means that half of bees, if I just cut everything in half. So instead of to the right and four up, I'm just gonna go one to the rights and to up like this. So this is gonna be my 10.5 b vector. And so my results in vector is just gonna be the shortest path from the start to the end. So this is going to be the result in vector here. So what we can see is that whenever you when you multiply a vector by a number that's greater than one, what you're gonna dio is you're gonna basically increase the magnitude or the length. So, for instance, this a here points in the same exact direction as this A. It's just twice as long. And then when you multiply a vector by number, that's less than one like we did for this one. It points in the same exact direction. It's just gonna be decreased in terms of the magnitude and the length. So what happens is and number that's greater than one makes the vector longer less than one makes the vector shorter. That's really all there is to it. So we could just calculate the resulting vector by you know, this is my see, we just count up the legs over here. And this is 1234567 This is 1234 So my hi pop news seven square post four squared and you get 8.6 Alright, guys, that's all there is to it. Let's go ahead and get another example. So we're gonna find the magnitude of this result in vectors, see? And it's gonna be 38 minus to be. So, by the way, all these rules work even for vector subtraction. Let's check it out. So we've got these two vectors here A and B, and basically all I have to do is we can think of this. Three A. Here is just being a plus a plus a just three vectors stacked on top of each other, lined up tip to tail and then this to be here. We're going to subtract B plus B. So let's go ahead and stack all those vectors together. We know that A is gonna be one to the right and one up, So that means three is just gonna be if I stack tomb or on top of those. So this whole entire thing here ends up being three a And so now we just have to add the negative to be well, my regular be vector is gonna be three to the right and one up. So from the tail of this one from the end of this one, I have to go. Not three the right and one up. I have to go in the opposite direction. I have to go through the left and then one down. So this is gonna be my negative be vector. So this is my negative B and I have to do it again. I have to do another three the left and then one down. So I'm gonna go this way. So notice how we've got this be vector here and now this vector is exactly twice as long. But now it just points in the opposite direction. That's all. That minus sign does. So now our result in vector here is gonna be from the start of the first to the end of the last one is basically if I just walked in this path, um and so you're gonna just draw the shortest path between those. This is my see vector, and then you just count up the legs. This is three and this is one. And so the magnitude which is given by the absolute value sign, is the square root of three squared plus one squared and you just get 3.16 and that is the magnitude. Alright, guys, that's all there is to it. Let me know if you have any questions.