Curves in space

A curve in space is described by a vector-valued function \(\mathbf{r}(t)=\langle x(t),y(t),z(t)\rangle\). Its velocity vector is the derivative \(\mathbf{v}(t)=\mathbf{r}'(t)=\langle x'(t),y'(t),z'(t)\rangle\). Speed is the magnitude of velocity: \(\|\mathbf{v}(t)\|=\sqrt{(x')^2+(y')^2+(z')^2}\). Acceleration is \(\mathbf{a}(t)=\mathbf{v}'(t)=\mathbf{r}''(t)\). To find the slope \(dy/dx\) on a plane curve, use \(\frac{dy}{dx}=\frac{y'(t)}{x'(t)}\) when \(x'(t)\neq 0\).