Vector Functions

A vector function is a function that is written as a vector. It looks like this:

$x\left(t\right)$ is the $x$-coordinate of the vector written as a function of $t$, and $y\left(t\right)$ is the $y$-coordinate of the vector also written as a function of $t$.

The Vector Functions

The position vector:

$\overrightarrow{r}\left(t\right)=\left(x\left(t\right),y\left(t\right)\right)$

The velocity vector:

$\overrightarrow{v}\left(t\right)={\overrightarrow{r}}^{\prime }\left(t\right)=\left(x^{\prime }\left(t\right),y^{\prime }\left(t\right)\right)$

The speed:

$\left|\overrightarrow{v}\left(t\right)\right|=\sqrt{x^{\prime }{\left(t\right)}^2+y^{\prime }{\left(t\right)}^2}$

The acceleration vector:

$\overrightarrow{a}\left(t\right)={\overrightarrow{v}}^{\prime }\left(t\right)={\overrightarrow{r}}^{″}\left(t\right)$

The acceleration:

$a=\left|\overrightarrow{a}\left(t\right)\right|=\sqrt{x^{″}{\left(t\right)}^2+y^{″}{\left(t\right)}^2}$