Multiplying a vector by a scalar
Let's say that I have the vector a, and let's say that it's equal to (2,1) So we could draw it right over here, So it's equal to (2,1), so if we were to start at the origin and we would move 2 in the horizontal direction, and 1 in the vertical direction so we would end up right over here. Now what I want to do is think about, how we can define multiplying this vector by a scalar for example, if I were to say 3 times the vector a. which is the same thing as saying 3 times (2,1) So 3 is just a number. One way to think about a scalar quantity, it is just a number, versus a vector is giving you how much you're moving in the various directions. It's giving you both the magnitude and a direction. while this is just a plain number right over here. But how do we define multiplying 3 times this vector? Well one reasonable thing that might jump out at you is, why don't we just multiply the 3 times each of these components? So this could be equal to.. we have 2 and 1.. And we're going to multiply each of these with 3. So 3 times 2 and 3 times 1. And then the resulting vector is still going to be a 2-dimensional vector. And it's going to be the 2-dimensional vector (6,3). Now I encourage you to get some graph paper out and to actually plot this vector, and think about how it relates to this vector right over here. So let me do that.. So the vector (6,3), if we started at the origin.. We would move 6 in the horizontal direction.. 1, 2, 3, 4, 5, 6.. And 3 in the vertical.. 1, 2, 3.. So it gets us right over there, so it would look like this. So what just happened to this vector? Well notice, one way to think about it is what has changed, and what has not changed about this vector? Well what's not changed is still pointing in the same direction. So this right over here has the same direction. Multiplying by the scalar, at least the way we defined it.. did not change the direction that my vector is going in. Or at least in this case it didn't.. But it did change its magnitude. Its magnitude is now 3 times longer, which makes sense! Because we multiplied it by 3. One way to think about it is we scaled it up by 3. The scalar scaled up the vector. That might make sense. Or it might make an intuition of where that word scalar came from. The scalar, when you multiply it, it scales up a vector. It Increased its magnitude by 3 without changing its direction. Well let's do something interesting.. Let's multiply our vector a by a negative number. Let's just multiply it by -1 for simplicity. So let's just multiply -1 times a. Well using the convention that we just came up with.. We would multiply each of the components by -1. So 2 times -1 is -2, and 1 times -1 is -1. So now -1 times a is going to be (-2,-1) So if we started at the origin, we would move in the horizontal direction -2, and in the vertical -1 So now what happened to the vector? When I did that? Well now it flipped its direction! Multiplying it by this -1, it flipped it's direction. Its magnitude actually has not changed, but its direction is now in the exact opposite direction. Which makes sense, that multiplying by a negative number would do that. In fact when we dealt with the traditional number line, that's what happened If you took 5 times -1, well now you're going in the other direction you're at -5, you're 5 to the left of zero. So it makes sense that this would flip its direction. So you could imagine, if you were to take something like -2 times your vector a, -2 times your vector a.. And I encourage you to pause this video and try this on your own.. What would this give? And what would be the resulting visualization of the vector? Well let's see, this would be equal to -2 times 2 is -4, -2 times 1 is -2, so this vector.. if you were to start at the origin! remember you don't have to start at the origin.. but if you were.. it would go 0, 1, 2, 3, 4.. 1, 2.. It looks just like this.. And so just to remind ourselves.. our original vector a looked like this.. (2, 1) looks like this.. And then when you multiply it by -2.. you get a vector that looks like this.. Let me draw it like this.. I'm purposely not having them all start at the origin, because they don't HAVE to all start at the origin.. But you get a vector that looks like this.. So what's the difference between a and -2 times a? well the negative flipped it over, and then the two flipped it over and now it has twice the magnitude but because of the negative it has twice the magnitude in the other direction.