Main content

### Course: Linear algebra > Unit 1

Lesson 1: Vectors- Vector intro for linear algebra
- Real coordinate spaces
- Adding vectors algebraically & graphically
- Multiplying a vector by a scalar
- Vector examples
- Scalar multiplication
- Unit vectors intro
- Unit vectors
- Add vectors
- Add vectors: magnitude & direction to component
- Parametric representations of lines

© 2024 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Multiplying a vector by a scalar

Watch Sal change the magnitude of a vector by multiplying it by a scalar. Created by Sal Khan.

## Want to join the conversation?

- Is it possible that you can do vectors in 3D?(14 votes)
- If you are asking if a vector can have 3 dimensions, then yes. It can even have more. V = [x, y, z](22 votes)

- How do you multiply vectors and negevative numbers?(5 votes)
- The same as you do with positive number except you flip the vector so it is going in the opposite direction. You essentially turn the vector 180 degrees.(18 votes)

- how can vectors be used?(6 votes)
- Vectors can be used in almost every branch of physics, and they have an application on almost everything of the real world. For example, vectors help classical mechanics to describe velocity, acceleration, force, or momentum. A car traveling to the east at x speed, a box sliding down a ramp, and a kid running to the farthest corner of a park. Vectors help more physics' branches such as electric and magnetic fields, quantum physics, and general relativity. For example, the force of a magnetic field and its direction, or a particle's position in a vector field.

http://physics.stackexchange.com/questions/14808/uses-of-vectors-in-real-life(9 votes)

- how can we define the matrix of vector which dont starts from the origin?(5 votes)
- I think that you are asking if there is a way to state the starting point for a vector. If that is the case, then I do not believe there is a convention for this. The matrix of the vector only states the motion of the vector from start point to end point. You would have to state the starting point separate from the vector matrix.(8 votes)

- At5:20you said the vector does not have to start at the origin. I was doing practice problems and I had a hard time figuring out what the answer would look like. The answers provided were not at the origin. It is easy to visualize the answer, i think, when the vector is at the origin. Why does it not have to start at the origin? Suggestions?(5 votes)
- Let's say you have to add two vectors and write the result in a matrix form (like Sal writes on the left). What options do you have (assuming you know the parallelogram rule of addition)?

I could think of 3, and only the last one got me to the right answer. Here, have a look: http://i.imgur.com/d5ZbsY0.png*Option 1*: They're not tail-to-tail. No idea how the sum vector would look like, right?*Option 2*: You found how the sum vector looks like, but you still can't find its coordinates (components) - they have to be measured relative to the origin! (A good question is... why? And the answer is, because people made up vectors and made up rules of adding them)*Option 3*: Now we've got it - if tails of both A and B are at (0;0), then we can easily find vector C's components and write it in matrix form. Yay!

Looking back at option 2, you could still, for example, find the length of vector C and a lot of other things. However, as you progress in linear algebra you're going to be a bit disappointed for the lack of graphical representations along the way. Linear algebra can work with N dimensions, not just two - and that's where its beauty lies.(7 votes)

- What's the bracket thing used to indicate vectors? Does it say anything about the direction the vector goes in?(2 votes)
- The 'bracket thing' is a matrix. The top number indicates the horizontal component of the vector and the bottom number the vertical component of the vector. Think of it as the distance in x and the distance in y.

so [2 1 ] (in columns) is 2 in x direction and 1 in the y direction.(10 votes)

- if a vector doesnt start at the origin but is multiplied by a negative scalar, then does the initial point still remain the same or is an inverted mirror vector formed?(5 votes)
- You can start the vector wherever you want. It doesn't matter where. You can keep the initial point or move it somewhere else.(4 votes)

- What are vectors used for? They don't seem very practical to me right now.(4 votes)
- Most of physics is vectors. Example: If you have a car going 60 mph to the north that clearly has both magnitude and direction. Now let's say a 40 mph wind pushes the cart East, after 2 hours how far is the car from its original starting point? You don't have to answer this question, but it'd be very difficult without the idea of vectors.(5 votes)

- How do you know what to multiply by?(5 votes)
- @1:28Sal multiples [2 1] by 3 because its the number he chose. Just like in any math problem, just multiply by what the problem asks you to.(3 votes)

- what happens, if we change the order of multiplication in vector product ?(3 votes)
- Scalar*Vector=Vector*Scalar,

Vector1 (dot) Vector2 = Vector2 (dot) Vector1,

This means that the commutative property applies to scalar multiplication, and the dot product, however:

Vector1 (cross) Vector2 does not equal Vector2 (cross) Vector1, it actually equals negative Vector2 (cross) Vector1(4 votes)

## Video transcript

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.