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Vector-Valued Function — Definition, Formula & Examples

A vector-valued function is a function whose input is a scalar (usually tt) and whose output is a vector. Each component of the output vector is itself a regular scalar function of tt.

A vector-valued function r:R→Rn\mathbf{r}: \mathbb{R} \to \mathbb{R}^n assigns to each scalar tt in its domain a vector r(t)=⟨f1(t),f2(t),…,fn(t)⟩\mathbf{r}(t) = \langle f_1(t), f_2(t), \dots, f_n(t) \rangle, where each fif_i is a real-valued function called a component function.

Key Formula

r(t)=⟨f(t), g(t), h(t)⟩=f(t) i+g(t) j+h(t) k\mathbf{r}(t) = \langle f(t),\, g(t),\, h(t) \rangle = f(t)\,\mathbf{i} + g(t)\,\mathbf{j} + h(t)\,\mathbf{k}
Where:
  • tt = Scalar parameter (often representing time)
  • f(t),g(t),h(t)f(t), g(t), h(t) = Component functions giving the x, y, and z coordinates
  • i,j,k\mathbf{i}, \mathbf{j}, \mathbf{k} = Standard unit vectors along the x, y, and z axes

How It Works

You evaluate a vector-valued function by plugging the parameter tt into each component function separately. The resulting vector traces out a curve in space as tt varies. In two dimensions, r(t)=⟨x(t),y(t)⟩\mathbf{r}(t) = \langle x(t), y(t) \rangle describes a plane curve; in three dimensions, r(t)=⟨x(t),y(t),z(t)⟩\mathbf{r}(t) = \langle x(t), y(t), z(t) \rangle describes a space curve. You can differentiate or integrate a vector-valued function by applying those operations to each component independently.

Worked Example

Problem: Given r(t)=⟨2t, t2, 3t−1⟩\mathbf{r}(t) = \langle 2t,\, t^2,\, 3t - 1 \rangle, find r(2)\mathbf{r}(2) and r′(t)\mathbf{r}'(t).
Evaluate at t = 2: Substitute t=2t = 2 into each component function.
r(2)=⟨2(2), (2)2, 3(2)−1⟩=⟨4, 4, 5⟩\mathbf{r}(2) = \langle 2(2),\, (2)^2,\, 3(2)-1 \rangle = \langle 4,\, 4,\, 5 \rangle
Differentiate each component: Take the derivative of each component with respect to tt.
r′(t)=⟨ddt(2t), ddt(t2), ddt(3t−1)⟩=⟨2, 2t, 3⟩\mathbf{r}'(t) = \langle \tfrac{d}{dt}(2t),\, \tfrac{d}{dt}(t^2),\, \tfrac{d}{dt}(3t-1) \rangle = \langle 2,\, 2t,\, 3 \rangle
Answer: r(2)=⟨4,4,5⟩\mathbf{r}(2) = \langle 4, 4, 5 \rangle and r′(t)=⟨2,2t,3⟩\mathbf{r}'(t) = \langle 2, 2t, 3 \rangle.

Why It Matters

Vector-valued functions are central to multivariable calculus (Calculus 3), where they describe curves and particle trajectories in space. Physics and engineering use them to model position, velocity, and acceleration of moving objects, making them essential for anyone studying mechanics or computer graphics.

Common Mistakes

Mistake: Confusing a vector-valued function with a multivariable scalar function like f(x,y)f(x, y).
Correction: A vector-valued function takes a scalar input and returns a vector. A multivariable function takes a vector input and returns a scalar. The directions of input and output are reversed.

Related Terms