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Chapter 10: Problem 63

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

Short Answer

Expert verified
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

01

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 \(g(t)\).
02

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 \(h(t)\).

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Most popular questions from this chapter

The position vector \(r\) describes the path of an object moving in space. Find the velocity, speed, and acceleration of the object. $$ \mathbf{r}(t)=t^{2} \mathbf{i}+t \mathbf{j}+2 t^{3 / 2} \mathbf{k} $$

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The definite integral of a vector-valued function is a real number.

In Exercises 39 and \(40,\) find the angle \(\theta\) between \(r(t)\) and \(r^{\prime}(t)\) as a function of \(t .\) Use a graphing utility to graph \(\theta(t) .\) Use the graph to find any extrema of the function. Find any values of \(t\) at which the vectors are orthogonal. $$ \mathbf{r}(t)=3 \sin t \mathbf{i}+4 \cos t \mathbf{j} $$

The position vector \(r\) describes the path of an object moving in the \(x y\) -plane. Sketch a graph of the path and sketch the velocity and acceleration vectors at the given point. $$ \mathbf{r}(t)=3 \cos t \mathbf{i}+2 \sin t \mathbf{j},(3,0) $$

The \(z\) -component of the derivative of the vector-valued function \(\mathbf{u}\) is 0 for \(t\) in the domain of the function. What does this information imply about the graph of \(\mathbf{u}\) ?
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