1. What a Vector-Valued Function Means
A vector-valued function is a function whose output is a vector instead of a single real number. The input is usually a real number, but the output has multiple components, like (x, y) or (x, y, z).
So instead of producing one value, the function produces a point in space.
2. Formal Definition
If a function sends each input to a vector in \( \mathbb{R}^n \), it is called a vector-valued function.
2.1. Definition
\vec{f} : A \to \mathbb{R}^n
\vec{f}(t) = (f_1(t), f_2(t), ..., f_n(t))
Each component function \( f_i(t) \) is a real-valued function.
3. Examples of Vector-Valued Functions
Here are simple examples that show how outputs become vectors:
3.1. Example 1 — 2D Output
\vec{f}(t) = (t, t^2)
Each input t gives a pair of values.
3.2. Example 2 — 3D Output
\vec{g}(t) = (t, \sin t, e^t)
Output is a point in 3D space.
3.3. Example 3 — From Ordered Pairs
A vector-valued function written explicitly:
\vec{h} = \{(1, (2,3)), (2, (5,7)), (4, (9,1))\}
Each input has a vector as the output.
4. Component Functions
A vector-valued function is just several real-valued functions combined together. Each part of the output comes from one of these smaller functions.
4.1. Illustration
\vec{f}(t) = (f_1(t), f_2(t))
f_1(t) = t^2, \quad f_2(t) = 3t + 1
5. Graphing Vector-Valued Functions
Graphing depends on the dimension of the output:
- 2D outputs → curves in the plane
- 3D outputs → curves in space
- Higher dimensions → cannot graph directly but can track components
6. Common Uses of Vector-Valued Functions
Vector-valued functions appear in many areas where motion or multidimensional outputs are natural.
6.1. Examples
- Motion of an object in 2D or 3D space
- Parametric curves
- Velocity and acceleration as vector outputs
- Transformations in geometry
7. Why Vector-Valued Functions Matter
They allow us to describe movement, geometry, and multidimensional patterns in a compact and organized way. By combining multiple real-valued functions into one vector output, they provide a powerful way to represent points, directions, and curves.