 Now let's talk about vectors that are equal and vectors that are opposite. If I've got two vectors that are equal to each other, they have to have the same length and they have to have the same direction. Opposite vectors also have the same length, but now they have to have opposite directions to each other. If we represent our vectors as arrows, this is pretty easy to see. Equal vectors are exactly the same length, point in the same direction in space. Opposite vectors are also the same length, but they're pointing in the opposite directions to each other. You can imagine if I took these two vectors and laid them on top of each other, they would exactly overlap. If I took these two vectors and laid them on top of each other, they'd be exactly the same length, but one's got a head on this side and the other one's got the head on the other side. If I represent my vectors as components, it's a little bit easier to see what's going on. If I've got two vectors, a and b, then their x components have to be exactly the same and their y components have to be exactly the same if I'm going to say those two vectors are equal to each other. It's really important here that all of the components must be equal. You could have two vectors that both have an x value of two, but if their y values aren't the same, then the two vectors aren't the same. If I've got opposite vectors, they've got the same numbers here, but they've got opposite signs. So where a has a positive sign, c has a negative sign. And the same thing on the y components. So if that's the case, then we can say that a is the minus of c, or it's the opposite of c. You could think of this if I multiplied a negative sign by the components of c, then I would have exactly the same. So again, all components have to be opposite. They have the same number, but they've got the opposite sign. Now if I represent my vectors as equations, then it's really easy to see that these two things are exactly equal to each other. The i's and the j's have to both match up. Same thing for opposite vectors, I have to have exactly the same numbers, but opposite signs on each and every term. Now so far, we've had nice integers for all of our things, but when I'm talking about vectors, it can be any number in there. So for example, this a and b are still the same because I still have exactly the same numbers and the same signs for each one of my components. And over here, a and c are still opposite of each other. I've got the same numbers in both cases, but now I've got opposite signs. In this case, a is positive and c is negative for the i component, whereas a is negative and c is positive for the j component. And if I were to multiply this c equation by a negative one, I've got exactly the same equation as a. As we're working with our vector equations in physics, it's important to keep track of what it means to be equal and what it means to be opposite.