A Deep Dive Into The World of Vector-Valued Function

General Form

If \( \mathbf{r}(t) \) is a vector-valued function, it can be written as:
\[
\mathbf{r}(t) = \langle x(t), y(t), z(t) \rangle
\]
Where \( x(t) \), \( y(t) \), and \( z(t) \) are scalar functions of \( t \), and \( \mathbf{r}(t) \) is a vector in 3-dimensional space. For a 2D vector-valued function, it might look like:
\[
\mathbf{r}(t) = \langle x(t), y(t) \rangle
\]
Here, the variable \( t \) (often representing time) is the parameter, and the components of the vector \( x(t) \), \( y(t) \), and \( z(t) \) are scalar functions of \( t \).

Example

Consider the vector-valued function for a circle:
\[
\mathbf{r}(t) = \langle \cos(t), \sin(t) \rangle
\]
As \( t \) varies, the vector \( \mathbf{r}(t) \) traces out a circle of radius \(1\) in the plane.

In 3D, a helix can be described by the vector-valued function:
\[
\mathbf{r}(t) = \langle \cos(t), \sin(t), t \rangle
\]
Here, the \( x \) and \( y \) components describe circular motion, while the \( z \) component increases linearly with \( t \), creating a helical shape.

Differentiation of Vector-Valued Functions

Just like with scalar functions, we can differentiate vector-valued functions. The derivative of a vector-valued function gives us the rate of change of the vector with respect to \( t \), often interpreted as the velocity vector in the context of motion.

To differentiate a vector-valued function \( \mathbf{r}(t) = \langle x(t), y(t), z(t) \rangle \), we differentiate each component function:
\[
\mathbf{r}'(t) = \langle x'(t), y'(t), z'(t) \rangle
\]

Example:

If \( \mathbf{r}(t) = \langle t^2, 3t, \sin(t) \rangle \), then:
\[
\mathbf{r}'(t) = \langle 2t, 3, \cos(t) \rangle
\]
This derivative gives the velocity of a particle at time \( t \).

Applications of Vector-Valued Functions

1. Physics and Motion: Vector-valued functions are commonly used to describe the motion of particles in space. The position of a particle at time \( t \) is given by a vector-valued function, and its velocity and acceleration can be found by differentiating that function.
2. Curves in Space: A vector-valued function can describe a curve in two or three dimensions. For example, the function \( \mathbf{r}(t) = \langle t, t^2, t^3 \rangle \) describes a curve in 3D space.
3. Engineering and Graphics: Vector-valued functions are also used in computer graphics, animation, and control systems to represent paths, trajectories, and object movements.

A vector-valued function is a function that outputs vectors, often used to describe curves, paths, or motions in 2D or 3D space. The components of the vector are scalar functions of one or more parameters (usually time), and you can differentiate and analyze these functions in a manner similar to scalar functions.