After 2 semesters of calculus concentrating on curves of the form y = f(x) , our next task is to extend what we have learned to the curves and surfaces which reside in 3 dimensional space. However, moving from one point on a line to another in 3 dimensions involves a rise up or down, a run forward or back, and a slide left or right. Slope only involves a rise and a run, thus making it strictly a 2 dimensional concept.
Thus, we begin this chapter by replacing the concept of slope with the concept of a vector. We then see that the study of curves in space is mathematically equivalent to the study of vectors and functions involving vectors. Indeed, the use of vectors allows us to introduce the fundamental concepts of velocity and acceleration in their true form, thus allowing us to study velocity and acceleration in their native form.
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