Vector Functions

A vector-valued function or vector function is a function from \( R^n \rightarrow R^k \) where \( k > 1 \) whose image is interpreted as a set of vectors. For example:

\[ r : R \rightarrow R^3 \] \[ r(t) = \langle f(t), g(t), h(t) \rangle \]

If we consider the image of \( r \) as position vectors, the tip of the moving vector \( r(t) \) traces out a space curve (or equivalently parametric curve):

\[ x = f(t), \ y = g(t), \ z = h(t) \]

Let us consider \( r(t) : R \rightarrow R^3 \). Now consider the derivatives of the analogous parametric equations:

\begin{itemize} \item The rate of change of \( x(t) \) with respect to \( t \): \( \textbf{x'(t)} \) \item The rate of change of \( y(t) \) with respect to \( t \): \( \textbf{y'(t)} \) \item The rate of change of \( z(t) \) with respect to \( t \): \( \textbf{z'(t)} \) \end{itemize}

From this, we can see that

\[ r'(t) = \langle x'(t), y'(t), z'(t) \rangle \]

Is the rate of change of \( r(t) \) with respect to \( t \). \( |r'(t)| \) is the speed at \( t \) if we interpret \( r(t) \) as a moving particle in space. If we consider \( r(t + h) - r(t) \) geometrically we can see that \( r'(t) \) is also the tangent vector to the curve traced out by the tips of \( r(t) \). Hence \( r'(t) \) works the same way as the derivative of 1 dimensional scalar functions.

The term "vector field" is often used synonymously with "vector function" but in general means \( n = k \).