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## Scalar multiplication

# Scalar multiplication: component form

## 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.