 So we've discussed the fact that lots of physical quantities have magnitude and direction. For example, displacement, velocity, acceleration, and Adele talked about forces. So all these different things that have a magnitude and a direction and they're called vectors. And they can be quite tricky to deal with because they're going in different directions. If you want to add them together then you need to do all sorts of trigonometry. Vanessa showed us a really cool way of dealing with that by breaking them down into components. For example, north and east. Or if you had three dimensions you might need north, east and up down. And we're going to look more closely now at vector notation in order to see how to deal with this in algebraic form. So what we're talking about here is that vector equations have all sorts of rules just like normal equations do. Let's look at one of the first vector equations we've seen. We've seen the definition of velocity. So velocity, which is a vector, is equal to the change in position over a change in time. Now that looks like just a number equals a number divided by another number. But we know that both velocity and position, these two things, have a direction. And so we needed to get the direction right. When we were doing all of this in one dimension, just along a line, we didn't worry about the direction. But when we do have to worry about the direction then we need to realize that the velocity is a vector. We just put the arrow on it to say that it's got a direction. And this change in position is also a vector, so it's got to have an arrow on it. So this is a vector equation. So vector equations have to obey all the same rules that normal equations do, plus at least one more. As an example, all vector equations must have the same units on the left and right hand side. So in our example here, our velocity has units of meters per second and we have meters divided by seconds. And so that matches units. The second rule is you can't have a vector equaling a scalar. So if I had some vector quantity, I don't even know what it is, A. And I said, oh, that's easy, that's just equal to B. Then you'd look at that and you'd say, alright, I know how big it is, but what direction is it? There isn't enough information in this equation here in order to figure out what direction A should be. And so that's not a valid equation. You can't have a vector equaling a scalar. And in exactly the same way, you can't have a scalar equaling a vector. So that's always wrong. That's basically the rules, but let's remind ourselves one more time what the equation means. For a vector equation, both the magnitude and the direction of the vectors on each side must be the same. And Vanessa showed us that this means that all the individual components of each vector have to be the same. So if you have one vector and you want to say that's the same as another vector, then your vector equation might look like that. And what we showed was that that means that the vector components in that direction are the same and the vector components in that direction are the same. So in other words, what we might say for our equation up here, for our velocity definition, we might say that the velocity in the north direction is equal to the change in position in the north direction divided by the time taken, and we'd also say that the velocity in the east direction is equal to the change in the position in the east direction divided by the time taken. So what that means is that one vector equation is equivalent to multiple scalar equations. So the velocity in the north direction is just a number, and this is just a number. The velocity in the east direction is just a number, and this is just a number. So these are scalar equations, and this one vector equation is equivalent to those two scalar equations. Or if we were talking about the vertical direction as well, it would be equivalent to three scalar equations. So when Adele told us about Newton's second law, in other words, that the sum of the forces acting on an object was equal to the mass times the acceleration of that object, that was actually a vector equation, and we know how to deal with that now. We could say that we know that the forces in the x direction are equal to the mass times the acceleration in the x direction, the forces in the y direction are equal to the mass times the acceleration in the y direction, and the net force in the z direction is equal to the mass times the acceleration in the z direction. One thing that's sort of buried in there is that the net force can be the combination of multiple forces. So we might have one force acting on an object, another force. We might have gravity and drag. We might have friction and normal force and gravity. We might have a spring acting on there. And so there must be some way in which we can add these things together. And again, Vanessa showed us the simple way to add these three forces together. In one sense, you've got to add them graphically, so you've got to sort of draw them and then figure out how to get from the start point to the end point, and that's lots of trigonometry. Or you could do it the easy way and you could just say, alright, the x component of that plus the x component of that plus the x component of that, give me the x component of that. And similarly, the y component plus the y component plus the y component, give me the y component, and the z plus the z plus the z, give me the z component. And then the x component of all those three things added together, give me the x component of that. So once we break a vector equation down into its components, it's just multiple equations of the type that we're used to. Now with this more nuanced understanding about vectors and how to do equations with them, you might look back at the previous things we've done and see equations like this. The equation for the force due to gravity is equal to the mass times acceleration due to gravity. Or you might see the simple model for the force due to friction where the frictional force is just equal to some constant times the normal force. Now you'll notice there are no arrows on these equations. We could put the arrows back on here because we know the force is a vector and the acceleration must be a vector. We say well we know that the force of gravity must be straight down, the acceleration must be straight down. So we could do that or we could say well we know what direction it is and all we were talking about here is the magnitude. So we know the magnitude of that force vector is equal to the mass times unknown magnitude of the acceleration due to gravity on Earth. Now in the frictional force case we definitely can't put the arrows on here because those things are not in the same direction. If we have a block on a surface and it's got a normal force stopping the block from falling through that surface, the frictional force is in a different direction. So this is definitely wrong. So that is definitely not an equation that is a vector equation, that is definitely an equation that is just talking about the magnitude of the frictional force compared to some constant times the magnitude of the normal force. So that is definitely a scalar equation. Another way of looking at this one back here is that it's the Z component, it's the downward component of the gravitational force and the downward component of the acceleration due to gravity. And the other two components happen to be zero. And so there are two possible things you could mean when you have an equation using things that are vectors with no arrows on top. And the first is it's just a representation of a single component of a vector equation and the second is that we're just talking about the magnitude of those vectors. And which is the correct one? Well, you have to get that from context. This is why it's very important to define your variables very carefully and the symbols that you use very carefully.