 Now we get to the third question on our worksheet. In the space below, draw a vector that is equal to 2 times d. And here's our d vector up here. I'm going to do the same sort of thing up here where I go through and say, reading this equation, the 4 i hat means that's 4 to the right because it's positive i. And the minus 2 means it's going to be 2 downwards. So that's what I'm expecting for d. I'm going to again just sort of drag this down so that we can talk a little bit more about it. When I've got 2d, I can either think of it as I'm going to go twice this d vector, or I can think about it as taking this d equation and going twice as far. So let me think about it in terms of drawing my vectors. So I'm going to start off here and I've got a vector which goes 4 to the right, 1, 2, 3, 4 and 2 downwards. And again, I'll just make that a little bolder so it's easier to see. That's vector d. Well, if I want a vector that's twice as long, I can make a copy of that and paste it over here. Sorry, it keeps jumping back and forth. So I've got d vector and another d vector, so it goes twice as far. Now if I were to draw this as just a single vector, I would want it to go that full distance as a single arrow. Now let's think about it in terms of the equation and the math. If I'm going to have 2d, it's like I've gone in and multiplied each one of these numbers times 2. Don't worry, I'll change my spacing so you can see it all here in a moment. And this spacing that I've got here shows us that I've got 2 times 4 and then 2 times 2. I could simplify that and that 2 times 4 is going to become 8 and the 2 times 2 is going to become 4. So my vector 2 times d is going to be 8i hat minus 4j hat. Or I have to go 8 to the right, 1, 2, 3, 4, 5, 6, 7, 8 and down 4, 1, 2, 3, 4. So this single long vector represents 2 times d and it's exactly equal to taking d twice. I just don't have everything lined up perfectly there. So this is what happens if you multiply a number times a vector. It means you've got a vector that's that much longer or you multiply each of the components by that amount.