 There's a second type of operation we can do with two vectors, which is called the dot product. Suppose I have two vectors u and v with the components u1 through un, v1 through vn. The dot product, which is also called the scalar product, is the sum of the component-wise products. In other words, we multiply the corresponding components and add them together. For example, suppose we wanted to find the dot product of the vector 3, 1, negative 2, 5 with 0, negative 1, 4, negative 3. So going back to our definition, we see the dot product is the sum of the component-wise products. So to find the dot product, we'll multiply the first components of the two vectors together, then the second components, the third, the fourth components, and we'll add them all together. Multiplying, then adding, gives us our dot product. It's worth taking a look at a special case of a vector dotted with itself. We might go back to the geometric idea of a vector as giving the directions from the origin to a terminal point where the coordinates of the terminal point correspond to the components of the vector. This means that the components of the vector will correspond to the lengths of two sides of the right triangle. But now that we have two sides of the right triangle, we can compute the length of the hypotenuse using the Pythagorean theorem. Now a useful strategy in mathematics is to try and get the same quantity in two completely different ways. So here we have the length of the hypotenuse using the Pythagorean theorem, but we might compare the length of the hypotenuse according to the Pythagorean theorem with the dot product of the vector with itself. And if we do that, we see that the dot product of a vector with itself will be the square of the length of the vector itself. And so this allows us to define a new quantity, which is called the modulus of the vector, which is defined to be the square root of the dot product of the vector with itself. For vectors in Rn, the modulus corresponds to the length of the vector. Since linear algebra is used by, well, really nearly everybody in nearly every quantitative field, the notation and terminology is inconsistent. So the modulus might also be called the two-norm of a vector, or it might be referred to as the magnitude of the vector. And because this notation for modulus looks a little too similar to our notation for absolute value, we might double those vertical bars to indicate that we actually want to talk about the modulus. Because you might run into any of these forms, we will make no attempt to be consistent in how we present our notation for the modulus or how we talk about it. There is, after all, strength in diversity. So let's find the magnitude of the vector u, 3, negative 1, 5, negative 6. And it's good to remember that the magnitude is the norm, is the modulus, is the square root of the dot product. So we'll go ahead and find the dot product. We'll multiply each component by itself and add up the products. And that gives us the magnitude of our vector.