Chapter 11: Vector Functions

Problem Zero: Read the section and make your own summary of the material.

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

$$\begin{gathered} \mathrm{Fill}\mathrm{in}\mathrm{the}\mathrm{blanks}\mathrm{to}\mathrm{complete}\mathrm{each}\mathrm{of}\mathrm{the}\mathrm{following}\mathrm{theorem}\mathrm{statements}: \\ \mathrm{The}\mathrm{derivative}\mathrm{of}\mathrm{a}\mathrm{vector}\mathrm{function}\mathrm{r}\left(\mathrm{t}\right)\mathrm{is}\mathrm{given}\mathrm{by}\mathrm{r}\text{'}\left(\mathrm{t}\right)=\underset{\mathrm{h}\to 0}{\lim }\mathrm{\_\_\_\_\_\_} \end{gathered}$$

Unit vectors: If v is a nonzero vector, explain why $\frac{\mathbf{v}}{\left|\left|v\right|\right|}$ is a unit vector.

Projecting one vector onto another: Show that the formula for the projection of a vector v onto a nonzero vector u is given by$pro{j}_{\mathit{u}}\mathit{v}\mathbf{}\mathbf{=}\mathbf{}\frac{\mathbf{u}\mathbf{.}\mathbf{v}}{\left|\left|u\right|\right|}\mathbf{,}\mathbf{}whereu\ne 0.$

Sketching vector functions: Sketch the following vector functions.

$\mathbf{r}\left(f\right)=\left\{t,t^3\right\},t\in \mathrm{ℝ}$

Let $r\left(t\right)=\left(x\left(t\right),y\left(t\right)\right),t\in [a,\infty ),$ be a vector-valued function, where a is a real number. Explain why the graph of r may or may not be contained in a circle centered at the origin. (Hint: Graph the functions $r_1\left(t\right)=\left(\frac{1}{t},\frac{1}{t}\right)$and $r_2\left(t\right)=\left(t,t\right),$both with domain [1,∞).)

Given a twice-differentiable vector-valued function $r\left(t\right)$, why does the principal unit normal vector $N\left(t\right)$ point into the curve?

Velocity and acceleration vectors: Find the velocity and acceleration vectors for the given vector functions.

$\mathbf{r}\left(t\right)=\left\{t,2t^2,3t^3\right\}$