Welcome back, everyone. So in the previous video, we talked about how you could add or subtract vectors using the tip to tail method. And what we're going to be talking about in this video is how you can multiply vectors by scalars. Now if you're not sure what a scalar is, don't sweat it, because it turns out all a scalar really is is a number that has magnitude but no direction. So an example of a scalar would be like 3 or 5 or negative 100, and you can multiply these scalars by vectors to get some interesting results. And that's what we're going to be talking about in this video. So without further ado, let's get right into things.
Now let's say you wanted to add 3 vectors together. If you wanted to do this, you could use the tip to tail method that we learned about in the previous video. So we have vector v, and I connect the tip of vector v with the tail of another vector v, and I connect this tip to tail with a third vector v, And this right here, drawing a resultant vector, would give us the vector v + v + v. So this is one way of solving this problem. But it turns out there's a more intuitive way that we could have added these vectors together. Rather than doing this three times, we could have just taken vector v and multiplied it by a scalar, 3. And that would, in theory, give us the same result as adding the vector to itself 3 times. Because whenever you multiply a vector by a scalar, it's either going to stretch or shrink your vector depending on what that scalar is. So I can see that our scalar is 3. And if we look at vector v, we can see vector v is 1, 2, 3 units to the right, and it's 1 unit up. So if I were to take both of these numbers here and multiply them by 3, that would give us the new vector. So 3 times 3 is 9, so we would go 9 units to the right, and then 3 times 1 is 3, so we would go 3 units up. So our vector would look like this if we multiplied vector v by 3. So this would be the vector 3 v, and notice how it's the same result that we got over here. The same magnitude, the same direction, but it was just a bit more simple since we multiplied v by a scalar instead. Now this process is actually going to become more intuitive as you do more examples of this.
So to make sure that we're really understanding how to multiply vectors by scalars, let's try some more complicated examples. So here we're told, given vectors u, v, and w, which we can see on this graph over here, sketch the resulted vector based on the operations below. And we're going to start with example a, which asks us to find one-half of vector u. Now I can see the scalar we're dealing with is 1/&2. So this is actually going to shrink our vector. Now I can see that our vector u is right there, so one-half of vector u would just be one-half of this. Now I'd say our vector u is 1, 2, 3, 4 units to the right and 1, 2 units up. So what I can do is shrink this by a half, which would be 2 units to the right and one unit up. So this would be one-half of vector u, and that is how you can solve example a. That's all there is to it. We found the answer.
Now let's try example b. Example b asks us to find a negative two times vector v. Now what I can see from this example is that we have a 2 being multiplied by our vector. So that's going to cause our vector to be stretched. Now I can see vector v is 2 units to the left and 3 units up. So what I'm going to do is scale this by 2. So that's going to cause our vector to be 1, 2, 3, 4 units to the left and 6 units up. So we would be somewhere up there. So vector v is going to look something like this, or vector 2 v, I should say. But I want you to notice something. This has a negative sign in front, and recall that negative signs in front of our vector reverses the direction. So rather than pointing up in this direction, it's actually going to point down here. It's going to oppose the direction that vector v points. So this would be the vector negative 2 v, and that is the answer to example b. See, it's pretty straightforward.
Now let's try example c. Example c asks us to find one-half of vector w + vector v. Now this is a bit more complicated because we first have to find the sum of vector v and w, and then we need to multiply this by 1/&2. We're going to take this by the steps. Now I can find w + v using the tip to tail method. So I see that this is w and the tip is right here and I'm going to connect this to the tail of v. So I'm going to draw v right there and this is going to be the vector w and vector v, and then drawing the resultant vector from the initial point of w to the terminal point of v. This is going to give us vector w + v. Now this is not the final result because this is not ultimately what we're looking for. We're looking for 1/&2 of w + v. And if I'm 1/&2 of w + v, well we just need to cut this vector in half and reduce it. So what I can see here is that the vector w + v is 1 unit down and 1, 2, 3, 4, 5, 6 units to the right. So half of that would be half a unit down and 3 units the righ...
