Vector Valued Function

Aka. vector function. A function whose domain is a set of real numbers and whose range is a set of vectors.

As $t$ varies, $\boldsymbol{r}(t)$ is positioned with its initial point at the origin, and its terminal point traces out $C$. In this case $\boldsymbol{r}(t)$ is the radius vector, or position vector, for $C$.

Component Function

$x(t)$, $y(t)$, $z(t)$ are the component functions of $\boldsymbol{r}(t)$.

Domain

Domain of $\boldsymbol{r}$ is the set of all $t$ for which $\boldsymbol{r}(t)$ is defined.

Space Curve

A vector-valued function $\boldsymbol{r}(t)$ whose values are three-dimensional vectors traces out a space curve in three-dimensional space.

Examples:

• $\boldsymbol{r}(t) = \langle \cos t, \sin t, 0 \rangle$ traces out the unit circle in the $xy$-plane centered at the origin.
• $\boldsymbol{r}(t) = \langle t, t^2, t^3 \rangle$ traces out the twisted cubic.