Vector-Valued Functions
Introduced| English |
|---|
| vector-valued function/ˈvektə ˈvæljuːd ˈfʌŋkʃn/ |
| vector/ˈvektə/ |
| parameter/pəˈræmɪtə/ |
A function that outputs an arrow
- Ordinary functions output a number; parametric ones output a pair of coordinates.
- A vector-valued function packages that pair into a single vector.
- Think of it as an arrow from the origin whose tip moves as time passes.
- It is the natural language for describing position, velocity, and motion in the plane.
What it outputs
- A vector-valued function 向量值函数 $\vec{r}(t)$ returns a vector 向量 for each input $t$.
- Usually that vector is a position: $\vec{r}(t) = \langle x(t), y(t)\rangle$.
- The parameter 参数 $t$ (often time) drives the arrow to a new spot.
- So one input gives one arrow, pointing from the origin to the current location.
A vector-valued function $\vec{r}(t)$ outputs, for each $t$, a…
A vector-valued function returns a vector for each input — often the position $\langle x(t), y(t)\rangle$.
The tip traces the path
- Hold the tail at the origin and let $t$ increase.
- The arrow's tip moves, and the trail it leaves is the curve.
- That curve is exactly the parametric path from earlier lessons.
- So a vector-valued function and a parametric function are two views of one idea.
A vector-valued function $\vec{r}(t) = \langle x(t), y(t)\rangle$ is closely related to a parametric function.
They are two views of the same idea: the components $x(t), y(t)$ are exactly the parametric equations.
As the parameter $t$ changes, the ____ of the position vector traces the curve.
The vector starts at the origin; its arrow tip marks the current point on the path.
For $\vec{r}(t) = \langle 2t,\ t^2\rangle$, what is the $y$-component at $t = 3$?
The $y$-component is $t^2 = 3^2 = 9$; the $x$-component is $2t = 6$, so $\vec{r}(3) = \langle 6, 9\rangle$.
Select all true statements about vector-valued functions.
They output a vector, not a single number. The other three are correct.
Evaluating and combining
- Evaluate $\vec{r}(t)$ by plugging $t$ into each component.
- $\vec{r}(3) = \langle 2\cdot 3,\ 3^2\rangle = \langle 6, 9\rangle$ for $\vec{r}(t) = \langle 2t, t^2\rangle$.
- The rate of change of $\vec{r}(t)$ is the velocity vector — components $\langle x'(t), y'(t)\rangle$.
- Add two vector-valued functions by adding their components.
Vector-valued functions
A vector-valued function outputs a vector; add the component vectors to get the resultant.
Why the vector view helps
- Writing motion as one vector keeps position, velocity, and direction together.
- It generalises cleanly to three dimensions, where you add a $z(t)$ component.
- Physics and computer graphics describe moving objects exactly this way.
- The single-arrow picture makes combined motions easy to reason about.
A vector-valued function outputs a vector, not a number. Do not treat $\vec{r}(t) = \langle x(t), y(t)\rangle$ as a single value — it always carries two components (a direction and a size), and both change with $t$.
A particle moves as $\vec{r}(t) = \langle t,\ t^2\rangle$.
- At $t = 0$: $\vec{r} = \langle 0, 0\rangle$ — it starts at the origin.
- At $t = 2$: $\vec{r} = \langle 2, 4\rangle$ — up and to the right.
- Its tip traces the parabola $y = x^2$ as $t$ increases.
A vector-valued function $\vec{r}(t)$ outputs a vector for each parameter value — usually the position $\langle x(t), y(t)\rangle$. Its tip traces a path as $t$ changes, so it is the vector view of a parametric function. Evaluate it component by component.