 Hello, and welcome to this screencast on section 9.3, the dot product. This screencast is going to cover the dot product, the angle between two vectors, and orthogonality. The dot product of two vectors, u and v, in n dimensions, is the sum of the product of the components. Note that the result of the dot product is a scalar, or real number. For example, consider these three dimensional vectors, u and v. To get the dot product of u and v, we are going to sum the product of the x components, with the product of the y components, with the product of the z components, and the result is the real number 3. Let's talk about some of the properties of the dot product. You can find a list of other properties in section 9.3 of your textbook. First, the dot product is commutative, which means that it doesn't matter what order we take the dot product of two vectors, it will still give us the same result. Also, the dot product gives us valuable geometric information about vectors. When we dot product a vector u with itself, we see that the result can be written in terms of the length of the vector u. And this holds, in general, the dot product of a vector with itself gives the square of the length of the vector. The dot product tells us more geometric information about vectors. Consider the angle between two vectors, denoted by theta in this picture. Using properties of the dot product, we can deduce the following useful relationship, which tells us that the dot product of two vectors u and v is equal to the product of the lengths of u and v times cosine of the angle between the two vectors. To see how this relationship is deduced, see section 9.3 of your textbook. Sometimes it is useful to think of this relationship as giving us the angle between two vectors, so we can rewrite it as we have here. On the next few slides, we will use this relationship to deduce other geometric information about vectors. The vectors u and v are orthogonal if the angle between them is a right angle. Note that cosine of a right angle is equal to zero, so using the relationship from the previous slide, we see that the dot product of two such vectors is equal to zero. Thus, we see that the dot product of two orthogonal vectors is zero. This gives us a quick way to check for orthogonality. Based on the values of cosine, we can gather more information from this relationship. If the angle theta is acute, then cosine of theta is positive. Therefore, the dot product of two such vectors must be positive. If theta is a right angle, as we saw in the previous slide, the dot product is zero. Lastly, if theta is obtuse, then cosine of theta is negative. Therefore, the dot product must also be negative.