 Now we're ready to move to the back side of the worksheet and you'll notice a couple of things right away First of all, there's no grid shown on here. So we're not going to be graphically representing our vectors We're just going to do some math with our vectors The other thing you'll notice pretty quickly as if you start looking at these I don't have integers anymore See vectors don't have to be integers We used integers when we're working with the grids because it's easy to count out three boxes in two boxes It's a little bit harder to count out 2.5 boxes or 1.7 boxes But in physics the type of things that we're going to represent with vectors aren't necessarily integers So we do need to know how to work with these things even if they're not integers We're going to take this one step at a time though So we're going to start with this first one an a plus b vector again And I've got 3.0 i hat plus 2.5 j hat minus 1.7 i hat minus 3.2 j hat if I use the same sort of Lining them up as columns as I did on the last one It looks something like this 3.0 i hat plus 2.5 j hat minus 1.7 i hat minus 3.2 j hat and We take care of the i's together. So we've got three minus 1.7 Now notice I'm not subtracting vector b vector b just has negative components and so when I do that Adding a negative number to a positive number. I get 1.3 i hat for the j for the i excuse me and minus 0.7 for the j Now it's probably easier for you to write this out in this column format and Certainly, that's perfectly acceptable But I want to show you one other way you might see it now Just so I've got some room. I'm going to temporary delete this you don't need to delete what's on your paper I'm going to grab another way of representing this and then walk you through what it says So I take my a vector here the full a vector put it in parentheses Plus my b vector and I put my full b vector Now just like an algebra if you had three a and four a you'd regroup them together We're going to regroup the i's together and we're going to regroup the j's together So I've got 3.0 minus 1.7 for the i and 2.5 minus 3.2 for the j and then when I simplify it I get the exact same answer I had before So it's often easier for students to see them if they stack them up in columns But you can also write it out as one long line You just have to keep track of the fact that you've got both i's and j's and you have to group the i's when you add them And group the j's So just as I a comparison here you could have both methods on your sheet Or you could just have one or the other of these methods on your sheet