 This lesson is going to go through some details on how to work with vectors because that's going to help us analyse displacement, velocity and acceleration and also we'll be able to understand and add together forces acting on objects. The most natural setting to start with is locations in space and since I'm drawing on a screen here I'm going to work with a two dimensional space. Let's look at a map of ANU. Now like most maps this one is drawn with north facing up the page and there's a little scale bar showing me roughly how far 200 metres is on this picture. My office is over by the lake here and shows us on the other side of campus. So if I go visit Joe for a chat I take this fairly indirect path but my overall displacement is given by the vector drawn in red. So if I had a programmable drone I could set it to fly 1300 metres at a bearing of 30 degrees east of north and it would get pretty close to Joe's office. So the 1300 metres is the magnitude of this vector and the 30 degrees east of north is its direction. Let's call this vector A. We write it either with a squiggle underneath or some people write it with an arrow over the top. Now I suppose Joe and I decide to go get a coffee. Our usual places are closed for the summer break so we head all the way over to the music school. Let's call the displacement from Joe's office to the music school the vector B. Then the overall displacement from my office to the music school is the vector A plus B or the purple arrow here. When Joe heads back to his office he's going to undergo a displacement that is the same distance, same length but in the opposite direction to the vector B and we're going to write that as minus B. Now each of these vectors can be defined by a magnitude which is their length and a compass bearing which defines their direction. And then the addition of vectors defined this way proceeds in the very geometrical graphical way I just drew them by joining the vectors up so the start point of the second vector is placed at the end of the first vector. But this is kind of unwieldy if you have to do it over and over again so I'm going to explain a much more straightforward approach now and it's all based on setting up a square grid or what mathematicians call an orthogonal coordinate system. So here's a grid instead of the detailed map and I've set it up so one box is 100 meters and the grid lines are running north south east west. Now imagine my programmable drone is only allowed to fly along the grid lines then to get from my office to Joe's office it has to fly 11 squares north and 7 squares to the east. It doesn't matter what order it flies these eastern north steps there will always be the same number in each of those two orthogonal directions. So I can write A as 1100 meters to the north plus 700 meters to the east and what I'm really doing here is I've broken down the vector A into the sum of two smaller vectors and the important property of these two vectors is that they are at right angles to each other that's what orthogonal means. Now the vector B points south east so it has a negative displacement in the north direction let's say it's negative 620 meters north plus 480 meters east and then if we come down to the purple vector we can count out the squares and see it's 480 meters to the north and 1180 meters to the east. The really great thing about turning all our vectors into sums of two parts that run along the same grid lines is that we just have to add the north south numbers and then the east west numbers there's no tricky drawing or measuring lines and angles and so just to check here you can see that the north part of A plus the north part of B really does equal the north part of A plus B. 1100 minus 620 is 480 and the same works for the east components of the vectors. I just need to make a final word about notation here I've used the symbols N and E to mean vectors that point in the directions north and east respectively but with just a single unit of length each.