Hey, guys. So up until now, we've seen how to simply add vectors together like a and b. But sometimes in problems, you're going to have to add multiples of vectors like 2a + 3b or 2a + 0.5b something like that. What we're going to see is that it works exactly like vector addition. The only thing that's different is that when you multiply a vector by sticking a number in front of it like a becomes 2a, then what's happening is because they have a number in front of it, the magnitude or the length is going to change, but not the direction. Let's check it out. So when we have vectors a and b, basically, you just line them up tip to tail like this, and your resultant vector is the shortest path from the start of the first to the end of the last. It's basically as 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 55 and so the magnitude would just be 52 + 52 and that's 7.07. What if instead of a + b, I was given 2a + 0.5b. Well, one way you can think about this, these multiples or these numbers in front is you can think about this 2a here as just being a+a. So you're still just doing vector addition here. This number here is basically kinda just like condensing all of this information, just into a single number. And these numbers are going to change the lengths of the vectors. So this is a + a and then you're going to add it to 0.5b, which you can think of as like half of a b. Let's check it out. So if my vector a is 3 to the right and one up, then that means that a + a or 2a is just going to be if I have 3 to the right and one up like this. And then you just add a tip to tail to another one, 3 to the right and one up like this. So we're going to have a and a And basically, this whole entire vector here is 2a. So then if I had to add it to 0.5b, we do the same thing. So my 0.5b Or sorry. My regular b vector is 2 to the right and 4 up. So that means that half of a b is if I just cut everything in half. So instead of 2 to the right and 4 up, I'm just going to go 1 to the right and 2 up like this. So this is going to be my 0.5b vector. And so my resultant vector is just going to 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 1, what you're going to do is you're going to 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 a number that's less than 1 like we did for this one, it points in the same exact direction. It's just going to be decreased in terms of the magnitude and the length. So what happens is a number that's greater than 1 makes the vector longer. Less than 1 makes the vector shorter. That's really all there is to it. So we can just calculate the resultant vector, by, you know, this is my c. We just count up the legs over here and this is 1, 2, 3, 4, 5, 6, 7. This is 1, 2, 3, 4. So my hypotenuse is 72 + 42 and you get 8.06. Alright guys. That's all there is to it. Let's go ahead and get another example. So we're going to find the magnitude of this resulting vector C and it's going to be 3A - 2B. So by the way, all these rules work even for vector subtraction. Let's check it out. So we've got these 2 vectors here, a and b. And basically, all I have to do is we can think of this 3a here as just being a + a + a. Just 3 vectors stacked on top of each other lined up tip to tail. And then this 2b here, we're going to subtract b+b. So let's go ahead and stack all those vectors together. We know that a is going to be 1 to the right and one up. So that means 3a is just going to be if I stack 2 more on top of those. So this whole entire thing here ends up being 3a. And so, now we just have to add the negative 2b. Well, my regular b vector is going to be 3 to the right and one up. So from the tail of this one, from the end of this one, I have to go not 3 to 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 going to be my negative b vector. So this is my negative b. And I have to do it again. I have to do another 3 to the left and then one down. So I'm going to go this way. So notice how we've got this b 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 resultant vector here is going to be from the start of the first to the end of the last one. It's basically as if I just walked in this path. And so you're going to just draw the shortest path between those. This is my c vector and then you just count up the legs. This is 3 and this is 1 and so the magnitude which is given by the absolute value sign is 32 + 12 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