The Tangent Vector to a Curve

Let C be a space curve parametrized by the differentiable vector-valued function r(t). At the point P_{0} (= r(t_{0})) of C, we have the derivative (or velocity) vector

v(t_{0}) = r′(t_{0}) = lim_{h→0} \frac{r(t_{0}+h) − r(t_{0})}{h} = lim_{h→0} \frac{\overrightarrow{P_{0}P}}{h},  (1)

where we write P for the point r(t_{0}+h) on C. We give a direct geometric interpretation of this vector, assuming it is not zero. It follows from equation (1) that for small h ≠ 0, we have P ≠ P_{0}, so that \overrightarrow{P_{0}P} is a nonzero vector.

Theorem 2 Assume that r′(t_{0}) ≠ 0. Let θ(h) be the angle between the vectors \overrightarrow{P_{0}P} and r′(t_{0}) (which is defined for sufficiently small h ≠ 0). Then:

(a) lim_{h→0+} θ(h) = 0;

(b) lim_{h→0−} θ(h) = π; or equivalently, the angle between \overrightarrow{P_{0}P} and −r′(t_{0}) tends to 0 as h → 0−.

Proof The standard angle formula gives

cos θ(h) = \frac{\overrightarrow{P_{0}P} • r′(t_{0})}{∥\overrightarrow{P_{0}P}∥ ∥r′(t_{0})∥} = \frac{[r(t_{0}+h) − r(t_{0})] • r′(t_{0})}{∥r(t_{0}+h) − r(t_{0})∥∥r′(t_{0})∥}

We cannot take the limit directly, because we get 0/0. However, if we first divide numerator and denominator by h and assume h > 0, we can apply equation (1) directly (using the continuity of norms and dot products) to get

cos θ(h) = \frac{\frac{r(t_{0}+h) − r(t_{0})}{h} • r′(t_{0})}{∥\frac{r(t_{0}+h) − r(t_{0})}{h}∥ ∥r′(t_{0})∥} ⟶ \frac{r′(t_{0}) • r′(t_{0})}{∥r′(t_{0})∥∥r′(t_{0})∥} = 1  (3)

as h → 0+. Finally, we apply the continuous function cos^{−1} to deduce that

θ(h) = cos^{−1}(cos θ(h)) ⟶ cos^{−1}(1) = 0 as h → 0+,

as required.

If h < 0, equation (3) is off by a sign, and we get θ → cos^{−1}(−1) = π instead. Geometrically, the vector r′(t_{0}) is tangent to the curve C at P_{0}. This leads to the

following definition.

Definition 4 The tangent line to C at P_{0} is the line through P_{0} in the direction of the vector r′(t_{0}).

Thus its parametric equation (with parameter u) is (see (13.3.2))

R(u) = r(t_{0}) + ur′(t_{0}).  (5)

110.202 Calculus III JMB File: curvtgt, Revision C; 11 Feb 2000; Page 1