 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, and I said, oh, that's easy, that's just equal to b. Then you'd look at that and you'd say, all right, 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. And 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. whilst 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 acceleration in the x direction, the forces in the y direction are equal to the mass times acceleration in the y direction, and the net force in the z direction is equal to the mass times 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 and 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 all right 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.