 Hello, and welcome to this screencast on section 9.3, the dot product. This screencast is going to cover work, force, and displacement, and projections. In physics, work is a measure of the energy required to apply a force to an object through a displacement. The picture here shows a force F displacing an object from point A to point B. The force F can be represented by a vector, since the force has both a magnitude and a direction. Similarly, the displacement of the object from A to B can be represented by the vector from A to B. Note that the force here is not necessarily parallel to the displacement of the object. In this situation, it turns out that the work required to displace the object is given by the dot product of these two vectors, and using the familiar relationship from earlier in this section, we see that the work done by the force is determined only by the magnitude of the force applied parallel to the displacement. The example from the previous slide shows that given two vectors u and v, sometimes we would like to be able to write the vector u as a sum of two vectors, one that is parallel to v, and the other perpendicular to v. These vectors are called projections. In particular, the vector that is parallel to v is called the projection of u onto v. Using properties of the dot product, we can deduce the following formula for the projection of u onto v. Note that the first term here is a real number. As we will see in our studies, it is sometimes useful to write the projection of u onto v as a scalar times a unit vector in the direction of v. This scalar is called the component of u in the direction of v.