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Scalar multiplication: component form

Sal explains what happens graphically and to the components of a vector when we multiply it by a scalar.

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Video transcript

- [Voiceover] What I wanna do in this video is give ourselves some practice, and hopefully some intuition, on multiplying a scalar times a vector. Now, what am I talking about when I say, multiplying a scalar times a vector? Well, let me set up a little two-dimensional vector here. Let's say I have the vector w, and let me give it an x component. Let's say its x component is one and its y component is, let's say it's two. And I could draw it if I like. Actually, that's a good idea. It's always nice to be able to visualize these things. Let me get some coordinate axis here. So, that's my x-axis, that is my y-axis. And so, if I were to draw this vector in standard form, I would put its initial point at the origin. And then its terminal point would be at the point one comma two. So its x coordinate is one, its y coordinate is going to be two. So, one, two. This vector is going to look like, its initial point is right here, its terminal point is going to be right over there, the vector, in standard, graphing it in standard form or visualizing it in standard form, would look like that. Now, of course, I could have the same vector and I could shift it around as long as I have the same length of the arrow and it's pointing in the same direction. But if no one tells you otherwise, it's nice to just put its initial point at the origin. Now, let's multiply it by a scalar. What do we mean by a scalar? Well, a vector is something that has a magnitude and a direction. A scalar is just something that has a magnitude. You could think of just the numbers that you started learning when you were four years old, those are scalars. So, for example, we could think about, what is three times w going to be? Three times w. Three is a scalar, w is a vector. Now, the convention we use for multiplying a scalar times a vector is, you just multiply each of the components times that scalar. So this is going to be equal to, we have a one and a two, and we're gonna multiply each of those times the three. Three times one, and then three times two, and so this is going to be equal to, this is going to be equal to, three times one is three, three times two is six. And so we see the resulting vector, we could call this vector three w, it's gonna have an x component of three and a y component of six. So if I were to draw it in standard form here, x component one, two, three, and then y component two, three, four, five and six. And so, it's going to look like this. Let me see if I can draw it reasonably. It's going to look like this. And obviously, I'm hand-drawing it, so it's not going to be exactly right. But it's going to look like that. So this is the vector three times w. Now, notice what happened when I multiplied it by the scalar. The direction didn't change, but the magnitude did. And you see what the magnitude changed by. It increased by a factor of three. The length of my blue arrow now is three times the length of my magenta arrow. Let's do another example. I'll use the same vector w, since we already have it set up. Let's multiply it times a negative scalar. Let's say, let's see what happens if I multiply negative two times w. And I (mumbles) a positive. You don't think about what this would be. And even, if you have the time, graph it out. Well, we would multiply each of our components by negative two. So it would be equal to negative two times one, would be the x component, and then the y component would be negative two times two. And so, this is going to be equal to the vector negative two comma negative four. Well, let's plot that. The x component is negative two. So it's negative one, negative two. And the y component, negative one, two, three, I'm going a little bit off of my axes, four, so that would be negative four there, that's negative two. So it's going to look something like this. It's going to look something like, something like that. So that right over there is the vector negative two w. Now let's think about what just happened. Well, because we had the negative here, it essentially flipped in the direction by 180 degrees. It's going in the opposite direction. But one way to think about it, they kind of would still sit on the same line. So the negative just flipped its direction. If you consider whatever direction this was, the magenta vector, w, was going, it's now going to go in the opposite direction. And then it also scaled it up by two. This is twice as long, has twice the magnitude of our original vector, and it's going in the opposite direction because of the negative sign. So hopefully this gives you a little bit on intuition of what it means to scale a vector. And literally, the word scalar, let me write it down. Scalar. That's a scalar, that's a scalar. It has the word scale in it. It's literally just scaling the vector. It is changing its magnitude. It might flip it around because of a negative sign, but it's essentially changing its magnitude, scaling it up or down or flipping it around with a negative sign.