Recall Alternative Formulas for Curvature, which states that the formula for the arc length of a curve defined by the parametric functions x = x ( t ), y = y ( t ), t 1 ≤ t ≤ t 2 x = x ( t ), y = y ( t ), t 1 ≤ t ≤ t 2 is given by We have seen how a vector-valued function describes a curve in either two or three dimensions. We explore each of these concepts in this section. This is described by the curvature of the function at that point. Or, suppose that the vector-valued function describes a road we are building and we want to determine how sharply the road curves at a given point. We would like to determine how far the particle has traveled over a given time interval, which can be described by the arc length of the path it follows. For example, suppose a vector-valued function describes the motion of a particle in space. In this section, we study formulas related to curves in both two and three dimensions, and see how they are related to various properties of the same curve. 3.3.3 Describe the meaning of the normal and binormal vectors of a curve in space. ![]() ![]() 3.3.2 Explain the meaning of the curvature of a curve in space and state its formula.3.3.1 Determine the length of a particle’s path in space by using the arc-length function.
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