Question

State the definition of a vector-valued function in the plane and in space.

Short answer

A vector-valued function in the plane maps each point in its domain to a vector in \(\mathbb{R}^{2}\) and is represented by a pair of functions. In space, it maps each point to a vector in \(\mathbb{R}^{3}\) and is represented by a triplet of functions.

Step-by-step solution

Definition in Plane

A vector-valued function in the plane is a function \(R: I\to\mathbb{R}^{2}\) where \(I\subseteq R\) (a subset of the real numbers representing an interval) and \(R(t) = \) where \(f\) and \(g\) are real-valued functions of the parameter \(t\). It's represented by the ordered pair of functions \(f(t)\) and \(g(t)\).

Definition in Space

A vector-valued function in the space is a function \(R: I\to\mathbb{R}^{3}\) where \(I\subseteq R\) (a subset of the real numbers representing an interval) and \(R(t) = \) where \(f\), \(g\), and \(h\) are real-valued functions of the parameter \(t\). It's represented by the ordered triplet of functions \(f(t)\), \(g(t)\), and \(h(t)\).