Vector Functions

A vector function is simply a rule that tells you where something is at each moment: put in a time, get back a position arrow. That's why it's called a vector function: its output is a vector instead of a single number. It's how you describe a flight path, a roller coaster track or a thrown ball. The notation $\langle f(t), g(t), h(t)\rangle$ looks heavy, but it's three ordinary one-variable functions side by side: $f(t)$ (how far east at time $t$), $g(t)$ (how far north) and $h(t)$ (how high). Pick a time, get a point. Let time run, get a curve.

The last lesson had functions with several inputs and one output. This one flips that around: one input, $t$, and three outputs. You've actually seen one already. The line $\vec r_0 + t\vec v$ from earlier in this unit is a vector function whose components happen to be linear. Swap in other functions and the path can bend:

• Lines: $\vec r_0 + t\vec v$.
• Circles: $\langle\cos t, \sin t, c\rangle$ goes around a circle at height $c$, since the first two components are the familiar unit circle and the height never changes.
• Helices: $\langle\cos t, \sin t, t\rangle$ does the same circling while the height keeps climbing, like a spring.
• The twisted cubic: $\langle t, t^2, t^3\rangle$.

Because everything happens one component at a time, the old rules carry over. The domain is where every component is defined, and limits are taken one component at a time.

Curves on surfaces. If the components satisfy a surface's equation for every $t$, the curve never leaves that surface. For example, $\langle\cos t, \sin t, t\rangle$ has $x^2 + y^2 = \cos^2 t + \sin^2 t = 1$ at every moment, so the helix winds around the cylinder $x^2 + y^2 = 1$.

Flight paths, roller-coaster tracks and the motion of a robot's gripper are vector functions; animators keyframe them and the computer fills in $\vec r(t)$ between.