Vector Functions
The Essentials
Vector function have the form:
\[
\vec{r}(t)=f(t)\hat{i}+g(t)\hat{j}+\ldots=\langle f(t),g(t),\ldots\rangle
\]
The limit definition of the derivative of a vector function is:
\[
\vec{r'}(t)=\lim_{\Delta t\to 0}\frac{\vec{r}(t+\Delta t)-\vec{r}(t)}{\Delta t}
\]
The following properties are true for the derivatives of vector functions:
\[
\text{Scalar Multiple: } \frac{d}{dt}(c\vec{r}(t))=c\vec{r'}(t)
\]
\[
\text{Sum and Difference: } \frac{d}{dt}(\vec{r}(t)\pm\vec{u}(t))=\vec{r'}(t)\pm\vec{u'}(t)
\]
\[
\text{Scalar Product: } \frac{d}{dt}(f(t)\vec{r}(t))=f'(t)\vec{r}+f(t)\vec{r'}(t)
\]
\[
\text{Dot Product: } \frac{d}{dt}(\vec{r}(t)\cdot\vec{u}(t))=\vec{r'}(t)\cdot\vec{u}+\vec{r}(t)\cdot\vec{u'}(t)
\]
\[
\text{Cross Product: } \frac{d}{dt}(\vec{r}(t)\times\vec{u}(t))=\vec{r'}(t)\times\vec{u}+\vec{r}(t)\times\vec{u'}(t)
\]
\[
\text{Chain Rule: } \frac{d}{dt}(\vec{r}(f(t)))=\vec{r'}(f(t))f'(t)
\]
If \( \vec{r}(t)\cdot\vec{r}(t)=c \) , then \( \vec{r}(t)\cdot\vec{r'}(t)=0 \)
The principle unit tangent vector is given by the equation:
\[
\vec{T}=\frac{\vec{r'}(t)}{||\vec{r'}(t)||}
\]