# Chapter 11: Vector Functions

Q. 0

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

Q. 1

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.

Q. 1

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

Q. 1

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

Q. 1

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

Q. 1

Sketching vector functions: Sketch the following vector functions.

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

Q. 10

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,∞)*.)

Q. 10

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?

Q. 10

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

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

Q. 10

$\mathrm{State}\mathrm{what}\mathrm{it}\mathrm{means}\mathrm{for}\mathrm{a}\mathrm{vector}\mathrm{function}\mathrm{r}\left(\mathrm{t}\right)=>">\mathrm{x}\left(\mathrm{t}\right),\mathrm{y}\left(\mathrm{t}\right),\mathrm{z}\left(\mathrm{t}\right)$