![]() ![]() 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\textk, when plotted versus x, the “area” under the line is just an algebraic combination of triangular “areas,” where “areas” above the x-axis are positive and those below are negative, as shown in (Figure). 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. ![]() 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. We can also construct negative infinitesimals, such as - 8 and -8. In words, you can express (Figure) for the work done by a force acting over a displacement as a product of one component acting parallel to the other component. The infinitesimals in R are of three kinds: positive, negative, and the real number 0. Which choice is more convenient depends on the situation. In two dimensions, these were the x– and y-components in Cartesian coordinates, or the r– and \phi -components in polar coordinates in three dimensions, it was just x-, y-, and z-components. ![]() We could equally well have expressed the dot product in terms of the various components introduced in Vectors. 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. ![]()
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