The Dot Product This is one way to multiply two vectors. The result is just a number, a scalar, which is why the dot product is also called 'scalar' product. When you multiply two vectors A and B, there are two ways to do it, depending on what is given: À· B = ABcosa where & is the angle between the two vectors. Or: ÷ B = A,B, + A, B, In this chapter the two vectors are the net force and the displacement. The sign of the work results directly from the cosine factor. We have two practice questions below. See the chapter on vectors for more on the dot product. In physics, work is done on an object when energy is transferred to the object. In other words, work is done when a force acts on something that undergoes a displacement from one position to another. Forces can vary as a function of position, and displacements can be along various paths between two points. We first define the increment of work dw done by a force acting through an infinitesimal displacement as the dot 17 product of these two vectors: dw = F .dr = F df | cos 0. Then, we can add up the contributions for infinitesimal displacements, along a path between two positions, to get the total work. Work Done by a Force The work done by a force is the integral of the force with respect to displacement along the path of the displacement: WAB = path AB F · d?. The vectors involved in the definition of the work done by a force acting on a particle are illustrated below. .B F dr path A. Vectors used to define work. The force acting on a particle and its infinitesimal displacement are shown at one point along the path between A and B. The infinitesimal work is the dot product of these two vectors; the total work is the integral of the dot product along the path. We choose to express the dot product in terms of the magnitudes of the vectors and the cosine of the angle between them, because the meaning of the dot product for work can be put into words more directly in terms of magnitudes and angles. In words, you can express the equation for the work done by a force acting over a displacement as a product of one component acting parallel to the other component. From the properties of vectors, it doesn't matter if you take the component of the force parallel to the displacement or the component of the displacement parallel to the force-you get the same result either way. Recall that the magnitude of a force times the cosine of the angle the force makes with a given direction is the component of the force in the given direction. The components of a vector can be positive, negative, or zero, depending on whether the angle between the vector and the component-direction is between 0" and