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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- Aren't numbers actually vectors because they have their absolute value as magnitude and a direction behind or ahead of zero?(32 votes)
- No, numbers by themselves are not vectors. Being behind or ahead of zero doesn't imply direction, unless you've explicitly started at zero. For example, 5 is 5 greater than zero. But it can also be 5 fewer than 10. So unless there's an origin and an end-point, you don't have a vector.
Besides, numbers can be just abstractions of quantities. If I have five apples, and the five is a vector, what direction does the five in my "five apples" show? Considering numbers to be automatically vectors leads to meaningless questions like that.(30 votes)
- If the scalar is a fraction, would the magnitude change?(9 votes)
- The magnitude would change. For example a vector with the magnitude (4,2) multiplied by the scalar (1/2) would have the magnitude (2,1).(15 votes)
- Multiply positive scalar = change magnitude, keep direction
Multiply negative scalar = change both magnitude and direction
Does this rule always apply when multiplying scalars to vectors, or is it only because he was particularly using -2 and 3?(7 votes)
- It's exactly analogous to multiplying negative and positive numbers. So in this case, yes.(10 votes)
- How is a scalar quantity able to change the direction when it itself only contains magnitude as shown at 4.47(4 votes)
- A scalar can only reverse the direction of a vector (i.e. by multiplying the vector by a negative scalar) but it cannot change the angle other than 180 degrees.(5 votes)
- How is the last vector the same if the direction has changed?(2 votes)
- It isn't, the product of the scalar ,-2 , and vector ,w , create a new vector. If the scalar is negative the direction of the new vector will be the opposite of the original vector though.(5 votes)
- can you use any scalar wile multiplying with a vector?(3 votes)
- Yes, any scalar can be used when multiplying with a vector.(2 votes)
- Since the instructor introduced the idea that we can change the magnitude by multiplying a vector with a positive scalar and we can change both the magnitude and the direction by multiplying a vector with a negative scalar, is it possible to just change the direction of a vector? If so, how (excluding multiplying by -1)?(2 votes)
- Yep, you would need to multiply by a matrix. For instance ultiplying the 2x2 matrix:
by a vector swaps the vector's x and y values. So multiplying that matrix by a vector like <2, 3> would make the vector <3,2> You do have to be a little careful witht he matrix to ensure the seze doesn't change.
If you did not learn about matrix multiplication yet you will eventually, or I could try and explain it.(1 vote)
- How did Sal get 3 for the scalar?(0 votes)
- It was just an example. He could've chosen 5 or 12 or -7, it was just to show how scalar multiplication works(4 votes)
- is there a way to multiply 2 vectors in the real plane? Or can we just do this in the complex plane?(1 vote)
- There are two standard ways to multiply vectors: the dot product, where the product of two vectors is a scalar, and the cross product, where the product of two vectors is another vector. Both are covered on Khan Academy.(2 votes)
- Can the direction of vectors be nonlinear?(1 vote)
- Yes. However, it is extremely complicated. I regret that I am not sure it is within my capabilities to show you how. But yes, it is possible.(1 vote)
- [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.