 So one of the most useful things we can compute given two vectors is known as the dot product. So I suppose I have two vectors u with components u1 through un and v with components v1 through vn. The dot product, sometimes called the scalar product, is defined by the following sum. Where we multiplied the first component of u with the first component of v, the second component of u with the second component of v, and so on, and added everything together. We can describe this as the sum of the component-wise products. For example, let's find the dot product of the vectors 3, 1, negative 2, 5, and 0, negative 1, 4, negative 3. So the dot product is the sum of the component-wise products, so I'll multiply the first components together. Multiply the second components together, multiply the third components together, multiply the fourth components together, and add them, and I get the dot product of negative 24. So what does the dot product tell us? It'll be useful to introduce some geometry. A vector in Rn is an ordered n-tuple whose components are elements of R. We might compare this vector in Rn to the coordinates of a point in Rn, which is an ordered n-tuple whose components are elements of R. And the fact that vectors and coordinates look so very much alike suggests the following line of reasoning. The coordinates of a point give the directions to the point from the origin, and this suggests that we might be able to interpret a vector as a set of directions for getting to a point from the origin. So what about the dot product? In R2, the vector u1, u2 tells us how to get to the point u1, u2 from the origin. The dot product itself is going to be u1 squared plus u2 squared. And so now we want to ask ourselves, self, where have we seen u1 squared plus u2 squared before? And if you remember that u1 and u2 as coordinates give us the horizontal and vertical distance to the point from the origin. From this, we see that the distance of the point from the origin can be found using the Pythagorean theorem, and that gives us that distance as square root of u1 squared plus u2 squared. And that looks a lot like the expression for the dot product of u with itself. In fact, this dot product is going to be the square of the distance of the point from the origin. Now if we're thinking of the vector as a set of directions for getting from the origin to the point, then the distance of the point from the origin is in some sense the length of the vector. And so this leads to the concept of the modulus of a vector written between these vertical bars. Again, there's only so many symbols, so these vertical bars don't mean absolute value. They don't mean determinant. They mean modulus. It's going to be defined as the square root of the dot product of the vector with itself. And so we might say that for vectors in Rn, the modulus corresponds to the length of the vector. Now because linear algebra is used by nearly everyone in nearly every quantitative field, notation and terminology tend to be a little inconsistent once you leave mathematics. So the modulus might also be called the two norm of a vector. And you might wonder if it's called a two norm. Is there a one norm, a three norm, a seventeen norm? And the answer is yes. We might also call the modulus the magnitude of the vector. And because it's easy to read the bars here as absolute value or determinant, sometimes we double them up to indicate the modulus. And if I were a kind and generous soul, I would take great pains to be consistent. But I'm not. More importantly, it's likely that you'll run into any one of these forms of notation or terminology. And so just to prepare you for the possibility that you might run into variant notation or variant terminology, we will make no effort to be consistent. And just as a quick example, let's find the magnitude of the given vector. And the magnitude is the norm, is the modulus, is the square root of the dot product. So we'll find the square root of the dot product. And we find then that the magnitude is equal to the square root of seventy-one.