 Since a vector represents a set of directions, we often represent vectors using boldface type, so v is a vector. Well, sometimes it's a little difficult to see that we have boldface type, so another thing we could do is we might draw an arrow over the name. So u with an arrow over it is a vector. The components of the vector are the values that correspond to the directions. So, for example, the vector 4 negative 3 has two components. The vector 1 4 3 has three components. The vector 2 negative 1 4 5 has, wait for it, four components. Now, this is a mathematical topic, and so the question we want to ask is, can we do arithmetic with vectors? And the answer is, we hope so, otherwise they won't be very useful. Let's introduce vector arithmetic this way. If a vector is a set of directions for getting from a point to another point, then the sum of two vectors should correspond to following both sets of directions, one after the other. So, for example, if I want to find the vector 3 5 plus the vector 1 4, then the vector 3 5 corresponds to the directions go right three units, then go up five units. Meanwhile, the vector 1 4 corresponds to the directions go right one unit, and then go up four units. And if we follow the directions one after the other, then the vector 3 5 plus 1 4 would tell us to go right three units and go up five units. Now, this next vector would tell us to go right one more unit, but we could just lump that together with the other rightward steps. So, we're going to go right one more unit, and then we're going to go up four more units. And so, all together, we should go right three plus one four units and up five plus four nine units. And this leads to the following idea. We can define vector addition as follows. Suppose U and V are two vectors, and we'll illustrate with two component vectors. Then the vector sum U plus V is going to be the sum of the first components, the sum of the second components, and similarly, with vectors with more components. And here, we'll introduce a useful phrase. Since the components of the sum are the sum of the components, we say that we have added component-wise. So, for example, let's say we want to add the vector 1 negative 3 5 to the vector 1 1 4. And so, we can add these algebraically, adding them component-wise. First component plus first component, that's 1 plus 1. Second component plus second component, that's negative 3 plus 1. And third components plus third component, 5 plus 4. We'll do the arithmetic. How about multiplication? There's actually two ways we can multiply vectors. The first comes from our definition of multiplication. And the definition that we started out with way, way, way, way, way back when you first started learning about multiplication was probably something like this. If m is a whole number, then mx is the sum of mx's. So if my vector is 1 4 5 negative 3, I can find 3v. 3v, well, that's the sum of 3v's. E equals means replaceable, so I know what v is as a vector. We know how to add two vectors, so I can add these first two vectors. I can add the third vector and get my vector sum. Now, the thing to notice here is that in the product mv, each component of v will be added to itself m times. And this is the same as multiplying each component by m. And so we can generalize this, which gives us one type of multiplication for vectors known as a scalar multiplication. For any real number c, cv is the vector produced by multiplying every component of our vector v by c. Now, you'll notice the real number c is the first factor that leads the vector. What about the vector times the real number c, vc? Well, my definition of multiplication suggests that the vc is going to be c plus c plus c, a whole bunch of c's added together, vector v times. It's not clear what this would mean. And so we'll employ a time-honored mathematical tradition. We'll leave it undefined for now. So let's find negative 3 times the vector 2, 5, 4. And so that will be the result of multiplying every component of our vector by negative 3, which gives us negative 6, negative 15, negative 12.