A parametric curve consists of the functions $$\{x(u),y(u),z(u)\}$$ together with an appropriate domain for the parameter $u$. The parameter $u$ determines both the direction and the speed that a given curve is traced out; a parametric curve is not merely a set of points. Consider for example a circle. You can go around it in either direction, and you can go around just once or many times. All of these are different parametric curves (can you write them down?) although the set of points involved (the graph) is the same in each case. We can rewrite this in vector form as \begin{equation} \{\rr(u):u\in[a,b]\} \end{equation} which is an example of a vector function (of one variable).

Here are some examples (without domains): \begin{eqnarray*} \hbox{circle}: \qquad && \rr(u) = a\cos u\xhat + a\sin u\yhat \cr \hbox{line}: \qquad && \rr(u) = \rr_0 + u\ww \cr y=f(x): \qquad && \rr(u) = u\xhat + f(u)\yhat \cr r=f(\theta): \qquad && \rr(\theta) = f(\theta)\cos\theta\xhat + f(\theta)\sin\theta\yhat \cr \end{eqnarray*}

Vector functions can be integrated and differentiated in the obvious way, using linearity. For instance, \begin{equation} {d\rr\over du} = {dx\over du}\xhat + {dy\over du}\yhat + {dz\over du}\zhat \end{equation} Thus, if $\vv=\vv(u)$ and $\ww=\ww(u)$ are vector functions, and writing primes to denote differentiation with respect to $u$, we have immediately that \begin{equation} (\vv+\ww)' = \vv' + \ww' \end{equation} In addition to linearity, the other main property of differentiation is the product rule. Since there are 3 products involving vectors (scalar multiplication, dot product, cross product), we get 3 product rules: \begin{eqnarray} (f\vv)' &=& f' \vv + f \vv' \cr (\vv \cdot \ww)' &=& \vv' \cdot \ww + \vv \cdot \ww' \cr (\vv \times \ww)' &=& \vv' \times \ww + \vv \times \ww' \cr \end{eqnarray} where $f=f(u)$.

The derivative $\rr'(u)$ is tangent to the parametric curve corresponding to $\rr(u)$. Thus, a vector equation for the tangent line to $\rr(u)$ at a given point $\rr(a)$ is simply $$\rr_T(u) = \rr(a) + u\rr'(a)$$ The unit tangent vector $\TT$ is defined by $$\TT = {\rr'\over |\rr'|}$$