Multivariate Calculus - Bruce E. Shapiro

Lecture 4
Lines and Curves in 3D
We can parameterize a curve in three dimensions in the same way that we did in
two dimensions, associating a function with each of the three coordinates,
x = f(t), y = g(t), z = h(t) (4.1)
where t is allowed to vary over some interval that we will call I. It is useful to
envision yourself as moving along the curve from one end to the other. At any time
t, you are at a point (x, y, z) on the curve. We think of the parameter t as time and
then
x = f(t) is your x-coordinate at time t
y = g(t)is your y-coordinate at time t
z = h(t) is your z-coordinate at time t.
Figure 4.1: A path can be parameterized by its coordinates.
We define our position vector r(t) at any time t as the vector pointing from
the origin to our position (x, y, z) at the time t. This vector is then a function of t
and is given by
r(t) = (x(t), y(t), z(t)) = f(t)i + g(t)j + h(t)k (4.2)
Suppose that we walk along a straight line, as illustrated in figure 4.3, starting at
the point P 0 , at a time t = 0, and arrive at the point P at time t, moving with a
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