If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content
Current time:0:00Total duration:5:44

Multiplying a vector by a scalar

Video transcript

let's say that I have the vector actually going to give it the vector a and let's say that it is equal to two one so we could draw it right over here so it's equal to 2/1 so in the horizontal if we were to start at the origin in the Hort we would move to in the horizontal direction and one 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 so for example if I were to say if I were to say three times three times the vector a which is the same thing as saying three times to one so three is just a number one way to think about a scalar quantity it is just a number versus a vector Y or this is giving you it's giving you how much you're moving in the various directions right over here it's giving you both a magnitude magnitude and a direction while this is just a plain number right over here but how would we how would we define multiplying three times this vector right over here well one reasonable thing that might jump out at you is well why don't we just multiply the three times each of these components so this could be equal to so we have two and one and we're going to multiply each of these times three so three times two and three times one and then the resulting vector is still going to be a two dimensional vector and it's going to be the two dimensional vector six three now encourage you to get some graph paper out in to actually plot this vector and think about how it relates to this vector right over this vector right over here so let me do that so the vector six three if we started at the origin we would move six in the horizontal direction one two three four five six and three of the vertical one two three so it gets us right over there so it would look it would look like this so what just happened to this vector well notice what one way to think about is what's changed and what has not changed about this enter well what's not changed is still pointing in the same direction so this right over here has the same 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 three times longer which makes sense because we multiplied it by three one way to think about it is we scaled it up by three the scalar scaled up the vector that might make sense or might give an intuition of where that word scalar came from the scalars can when you multiply it it scales up it scales up with AK vector it increased its magnitude by three without changing its direction but let's do something interesting let's multiply our vector a let's now multiply it by a negative number let's actually just multiply it by negative one just for simplicity so let's just multiply negative one negative one times a well using the convention that we just came up with we would multiply each of the components by negative one so 2 times negative 1 is negative 2 and 1 times negative 1 is negative 1 so now negative 1 times a is going to be negative 2 negative 1 so if we start at the origin we would move negative we would move in the horizontal direction negative 2 and in the vertical direction negative 1 so now what happened to the vector now what happened to the vector when I did that well now it flipped its direction multiplying it by this negative 1 it flipped its direction its magnitude it 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 just dealt with the traditional number line that's what happened if you took 5 times negative 1 well now you're going in the other direction you're at negative 5 your five 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 negative 2 times your vector a negative 2 times your vector 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 negative 2 times 2 is negative 4 negative 2 times 1 is negative 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 be so you'd go 0 1 2 3 4 1 2 it looks just like this it looks just like this and so just to remind ourselves our original vector a our original vector a look like this our original vector a look like this two one looks like this and then when you multiply it by negative two you get a vector that looks like this you get a vector that looks like this let me draw it like this and I'm and 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 that looks like this so what's the difference between a and negative 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