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Scalar multiplication: magnitude and direction

Sal analyzes the magnitude and direction of vectors that are a result of scalar multiplication of a vector whose magnitude is given.

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  • duskpin sapling style avatar for user Ishy Ish
    At Sal said that the magnitude can only be positive. But shouldn't that be non-negative?
    (36 votes)
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  • purple pi purple style avatar for user alex2002park
    what if the x and y have different multiples? Like (5x,8y)?
    (12 votes)
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    • starky sapling style avatar for user aa96sk
      No ones replied to you so I thought i'd give it a go...
      Basically, you're not multiplying the vector by a scalar. You're multiplying its direction components by two separate scalars, if you want to think of it like that. It's the same principle however, you would just do √(5x)²+(8y)² ... we can't really imagine what this would look like however, as we don't know what the initial x,y components of the vector looks like. Although if you'd like an example:
      Say vector 'V' has the direction components 3i + 4j then the magnitude of the vector would be √(3)²+(4)² which would equal 5. When applying your multiples (5 and 8) to each of the x, y components it would become √(15)²+(32)² which gives you 35.34 roughly...
      So as you can see when multiplying the components by separate variables, it's no longer a simple scalar multiplication of the vector where we can just multiply the initial magnitude by say 3. Hope this clears things up.
      (12 votes)
  • aqualine ultimate style avatar for user Vincent Lin
    If a vector can have zero length, what would be its direction?
    (8 votes)
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  • mr pants teal style avatar for user Dennis.qx.Xiao
    how would you solve it if lets say that vector z was (-2x,3y)?
    (4 votes)
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  • piceratops ultimate style avatar for user Luca Vernhes
    For the second part of the first question, shouldn't the vector of z equal -10 not 10?
    (2 votes)
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  • starky seedling style avatar for user Roumel Rahman
    why does the vector have to be always with a letter and not a number?
    (2 votes)
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  • blobby green style avatar for user 22aggarwaln
    magnitude of a vector is √x^2 + y^2
    So, √x^2 + y^2 = 5
    x^2 + y^2 = 25
    3(x^2 + y^2) = 75
    3x^2 +3y^2 = 75
    √3x^2 +3y^2 = √75
    ans = 5√3 = 8.66
    What am i doing wrong here?
    (1 vote)
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    • leaf green style avatar for user kubleeka
      If you scale a vector by 3, <x, y> becomes <3x, 3y>. x is replaced by 3x, y is replaced by 3y.

      So if the magnitude is √(x²+y²)=5, we replace x by 3x and y by 3y to get
      √((3x)²+(3y)²)
      =√(9x²+9y²)
      =√(9(x²+y²))
      =√9√(x²+y²)
      =3√(x²+y²)
      =3·5=15.
      (4 votes)
  • leaf red style avatar for user Ed
    At the very beginning, since the magnitude of vector v is equal to 5, could you say that (x,y) would be (5x, 5y)? Thanks.
    (1 vote)
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    • purple pi purple style avatar for user doctorfoxphd
      No, because the magnitude of a vector is equal to the distance from the beginning point to the end point. The components x and y are the individual contributions in the x direction and y direction that would push the beginning point location to the end location. So the components x and y would be the sides of a right triangle and the magnitude of 5 is the length of the arrow (vector) that forms the hypotenuse. There are pretty much an infinite number of ways to get 5 as the magnitude. We can use trig to help us, but in your case, we can use x² + y² = v² from Pythagoras to figure out the components.
      However, after we have said that the magnitude is 5 if the components are x and y, we cannot multiply the components by 5 and have the same vector. That would be a new, bigger vector.
      (3 votes)
  • spunky sam blue style avatar for user Adnan Nakib
    If a vectors x and y components are multiplied by different scaler numbers then, how to find the magnitude?
    (2 votes)
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  • old spice man green style avatar for user Shazib Howlader
    While we write the magnitude of a vector, why we use the double bar (//v//)? Could not we use the single bar(/v/)?
    (1 vote)
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Video transcript

- [Voiceover] All right, so we're told that vector v is equal to x comma y. They're not giving us the actual numbers for the x component and the y component. They're just saying that the x component is going to be the variable x. And the y component is going to be the variable y. And the magnitude of vector v is going to be equal to five. All right, I think I can digest all of that. And it says, fill in the blanks to complete the following sentences. So they're saying vector w right over here, they didn't have to actually write the word vector. They have an arrow on top of the w, so we know it's a vector. So vector w its components are three x and three y. So its x component is three times the x component of vector v. And its y component is three times the y component of vector v. Instead of having a y, it has a three y. Instead of having an x, it has a three x. So w is essentially vector v being multiplied by the scalar three. So its magnitude is going to be three times the magnitude of vector v. So if vector v has a magnitude of five, vector w's going to have a magnitude of 15. It's going to have three times the magnitude. All right, now they tell us vector z is equal to, so its x component is negative two times the x component of vector v. And its y component is negative two times the y component of vector v. So you could view vector z as negative two times vector v. Or the scalar negative two times the vector v. And so you might say okay that means its magnitude is going to be negative two times the magnitude of v. But you have to remember, magnitude, that's always going to be positive. One way to think, when we talk about even absolute value we're really thinking about the magnitude of something. We're talking about how long it is. We're not thinking about its direction. So we wouldn't think about the negative there. If we think about its direction, this negative is going to flip its direction and we could think about that in a second. But if we just think about magnitude, we care about the two. So its going to have two times the magnitude of vector v. So instead of five, it's going to be five times two, which is ten. So now they say, match each vector to defined to a vector that could be its equivalent. So if we think about it, the vector with the smallest magnitude is going to be vector v. The other two we're multiplying vector v by scalars with an absolute value of more than one. So they're going to have larger magnitudes. So the smallest magnitude is this one right over here. This is the shortest vector. So that would be our vector v. Now, vector w would be one that has three times the magnitude and it's going in the same direction. So, this one has three times the magnitude. It looks like three times, roughly. And it's definitely going in the same direction. And then vector z would have two times the magnitude. But since we have, we're multiplying it by the negative scalar, it's going to go in the opposite direction. And this looks like it. This purple arrow looks like it's about twice the length of this red arrow. And it's going in the opposite direction. So I feel pretty good about what we have done here. Let's do one more of these. So, let's do this one here. Here we have a vector v again. It has a magnitude of 10. And it says, fill in the blanks to complete the following sentences. Vector w now, so now vector w, it's taking each of the components of vector v and multiplying them by one-fifth. So you can view vector w as being equal to one-fifth, the scalar one-fifth times vector v. The magnitude of w is going to be one-fifth the magnitude of v. One-fifth of 10 is just going to be two. Now vector z, it looks like it's three-fifths times vector v. We're multiplying each of the components times three-fifths. So that's like multiplying the entire vector by the scalar three-fifths. So it's magnitude is going to be three-fifths the magnitude of vector v. And so three-fifths of 10 is six. All right, then they say, match each vector defined above to a vector that could be its equivalent. So here the vector with the largest magnitude is vector v. This would be vector v. And the vector with the smallest magnitude is vector w. It's one-fifth the magnitude. So this has a magnitude, should be 10. This should one one-fifth of that which is a magnitude of two and it looks like that. And this should be three-fifths of the 10, which is a magnitude of six. And that looks about right, so I feel good with what we have filled out. So hopefully that gives you a sense of how to tackle these types of example problems.