Our Discord hit 10K members! 🎉 Meet students and ask top educators your questions.Join Here!

Chapter 10

VECTORS AND THE GEOMETRY OF SPACE

Educators


Problem 1

Find the length of the curve.
$$\mathbf{r}(t)=\langle t, 3 \cos t, 3 \sin t\rangle, \quad-5 \leqslant t \leqslant 5$$

Anna Marie V.
Numerade Educator

Problem 2

Find the length of the curve.
$$\mathbf{r}(t)=\cos t \mathbf{i}+\sin t \mathbf{j}+\ln \cos t \mathbf{k}, \quad 0 \leqslant t \leqslant \pi / 4$$

Anna Marie V.
Numerade Educator

Problem 3

Find the length of the curve.
$$\mathbf{r}(t)=\mathbf{i}+t^{2} \mathbf{j}+t^{3} \mathbf{k}, \quad 0 \leq t \leqslant 1$$

Anna Marie V.
Numerade Educator

Problem 4

Find the length of the curve.
$$\mathbf{r}(t)=12 t \mathbf{i}+8 t^{3 / 2} \mathbf{j}+3 t^{2} \mathbf{k}, \quad 0 \leqslant t \leqslant 1$$

Anna Marie V.
Numerade Educator

Problem 5

Find the length of the curve correct to four decimal places, (Use your calculator to approximate the integral.)
$$\mathbf{r}(t)=\left\langle t^{2}, t^{3}, t^{4}\right\rangle, \quad 0 \leq t \leqslant 2$$

Anna Marie V.
Numerade Educator

Problem 6

Find the length of the curve correct to four decimal places, (Use your calculator to approximate the integral.)
$$\mathbf{r}(t)=\left\langle t, e^{-t}, t e^{-t}\right\rangle, \quad 1 \leq t \leqslant 3$$

Anna Marie V.
Numerade Educator

Problem 7

Let $C$ be the curve of intersection of the parabolic cylinder
$x^{2}=2 y$ and the surface $3 z=x y .$ Find the exact length of
$C$ from the origin to the point $(6,18,36)$ .

Anna Marie V.
Numerade Educator

Problem 8

Graph the curve with parametric equations $x=\cos t$
$y=\sin 3 t, z=\sin t .$ Find the total length of this curve
correct to four decimal places.

Anna Marie V.
Numerade Educator

Problem 9

Reparametrize the curve with respect to arc length measured from the point where $t=0$ in the direction of increasing $t .$
$$\mathbf{r}(t)=2 t \mathbf{i}+(1-3 t) \mathbf{j}+(5+4 t) \mathbf{k}$$

Anna Marie V.
Numerade Educator

Problem 10

Reparametrize the curve with respect to arc length measured from the point where $t=0$ in the direction of increasing $t .$
$$\mathbf{r}(t)=e^{2 t} \cos 2 t \mathbf{i}+2 \mathbf{j}+e^{2 t} \sin 2 t \mathbf{k}$$

Anna Marie V.
Numerade Educator

Problem 11

Suppose you start at the point $(0,0,3)$ and move 5 units
along the curve $x=3 \sin t, y=4 t, z=3 \cos t$ in the positive direction. Where are you now?

Anna Marie V.
Numerade Educator

Problem 12

Reparametrize the curve
$$\mathbf{r}(t)=\left(\frac{2}{t^{2}+1}-1\right) \mathbf{i}+\frac{2 t}{t^{2}+1} \mathbf{j}$$
with respect to arc length measured from the point $(1,0)$ in
the direction of increasing $t$ . Express the reparametrization
in its simplest form. What can you conclude about the
curve?

Anna Marie V.
Numerade Educator

Problem 13

(a) Find the unit tangent and unit normal vectors $\mathbf{T}(t)$ and $\mathbf{N}(t)$ .
(b) Use Formula 9 to find the curvature.
$$\mathbf{r}(t)=\langle t, 3 \cos t, 3 \sin t\rangle$$

Anna Marie V.
Numerade Educator

Problem 14

(a) Find the unit tangent and unit normal vectors $\mathbf{T}(t)$ and $\mathbf{N}(t)$ .
(b) Use Formula 9 to find the curvature.
$$\mathbf{r}(t)=\left\langle t^{2}, \sin t-t \cos t, \cos t+t \sin t\right\rangle, \quad t>0$$

Anna Marie V.
Numerade Educator

Problem 15

(a) Find the unit tangent and unit normal vectors $\mathbf{T}(t)$ and $\mathbf{N}(t)$ .
(b) Use Formula 9 to find the curvature.
$$\mathbf{r}(t)=\left\langle\sqrt{2} t, e^{t}, e^{-t}\right\rangle$$

Check back soon!

Problem 16

(a) Find the unit tangent and unit normal vectors $\mathbf{T}(t)$ and $\mathbf{N}(t)$ .
(b) Use Formula 9 to find the curvature.
$$\mathbf{r}(t)=\left\langle t, \frac{1}{2} t^{2}, t^{2}\right\rangle$$

Anna Marie V.
Numerade Educator

Problem 17

Use Theorem 10 to find the curvature.
$$\mathbf{r}(t)=t^{3} \mathbf{j}+t^{2} \mathbf{k}$$

Anna Marie V.
Numerade Educator

Problem 18

Use Theorem 10 to find the curvature.
$$\mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}+e^{\prime} \mathbf{k}$$

Anna Marie V.
Numerade Educator

Problem 19

Use Theorem 10 to find the curvature.
$$\mathbf{r}(t)=3 t \mathbf{i}+4 \sin t \mathbf{j}+4 \cos t \mathbf{k}$$

Anna Marie V.
Numerade Educator

Problem 20

Find the curvature of $\mathbf{r}(t)=\left\langle t^{2}, \ln t, t \ln t\right\rangle$ at the
point $(1,0,0) .$

Anna Marie V.
Numerade Educator

Problem 21

Find the curvature of $\mathbf{r}(t)=\left\langle t, t^{2}, t^{3}\right\rangle$ at the point
$(1,1,1) .$

Anna Marie V.
Numerade Educator

Problem 22

Graph the curve with parametric equations $x=\cos t$
$y=\sin t, z=\sin 5 t$ and find the curvature at the
point $(1,0,0)$

Check back soon!

Problem 23

Use Formula 11 to find the curvature.
$$y=x^{4}$$

Anna Marie V.
Numerade Educator

Problem 24

Use Formula 11 to find the curvature.
$$y=\tan x$$

Anna Marie V.
Numerade Educator

Problem 25

Use Formula 11 to find the curvature.
$$y=x e^{x}$$

Anna Marie V.
Numerade Educator

Problem 26

At what point does the curve have maximum curvature? What happens to the curvature as $x \rightarrow \infty ?$
$$y=\ln x$$

Anna Marie V.
Numerade Educator

Problem 27

At what point does the curve have maximum curvature? What happens to the curvature as $x \rightarrow \infty ?$
$$y=e^{x}$$

Anna Marie V.
Numerade Educator

Problem 28

Find an equation of a parabola that has curvature 4 at the
origin.

Check back soon!

Problem 29

(a) Is the curvature of the curve $C$ shown in the figure
greater at $P$ or at $Q ?$ Explain.
(b) Estimate the curvature at $P$ and at $Q$ by sketching the
osculating circles at those points.

Check back soon!

Problem 30

Use a graphing calculator or computer to graph both the curve and its curvature function $\kappa(x)$ on the same screen. Is the graph of $\kappa$ what you would expect?
$$y=x^{4}-2 x^{2}$$

Anna Marie V.
Numerade Educator

Problem 31

Use a graphing calculator or computer to graph both the curve and its curvature function $\kappa(x)$ on the same screen. Is the graph of $\kappa$ what you would expect?
$$y=x^{-2}$$

Anna Marie V.
Numerade Educator

Problem 32

Plot the space curve and its curvature function $\kappa(t)$.
Comment on how the curvature reflects the shape of the
curve.
$$\mathbf{r}(t)=\langle t-\sin t, 1-\cos t, 4 \cos (t / 2)\rangle, \quad 0 \leqslant t \leqslant 8 \pi$$

Check back soon!

Problem 33

Plot the space curve and its curvature function $\kappa(t)$.
Comment on how the curvature reflects the shape of the
curve.
$$\mathbf{r}(t)=\left\langle t e^{t}, e^{-t}, \sqrt{2} t\right\rangle, \quad-5 \leqslant t \leqslant 5$$

Check back soon!

Problem 34

Two graphs, $a$ and $b,$ are shown. One is a curve $y=f(x)$ and the other is the graph of its curvature function $y=\kappa(x) .$ Identify each curve and explain your choices.

Anna Marie V.
Numerade Educator

Problem 35

Two graphs, $a$ and $b,$ are shown. One is a curve $y=f(x)$ and the other is the graph of its curvature function $y=\kappa(x) .$ Identify each curve and explain your choices.

Anna Marie V.
Numerade Educator

Problem 36

Use Theorem 10 to show that the curvature of a plane
parametric curve $x=f(t), y=g(t)$ is
$$\kappa=\frac{|\dot{x} \ddot{y}-\dot{y} \ddot{x}|}{\left[\dot{x}^{2}+\dot{y}^{2}\right]^{3 / 2}}$$
where the dots indicate derivatives with respect to $t$.

Anna Marie V.
Numerade Educator

Problem 37

Use the formula in Exercise 36 to find the curvature.
$$x=e^{t} \cos t, \quad y=e^{t} \sin t$$

Anna Marie V.
Numerade Educator

Problem 38

Use the formula in Exercise 36 to find the curvature.
$$x=a \cos \omega t, \quad y=b \sin \omega t$$

Anna Marie V.
Numerade Educator

Problem 39

Find the vectors $\mathbf{T}, \mathbf{N},$ and $\mathbf{B}$ at the given point.
$$\mathbf{r}(t)=\left\langle t^{2}, \frac{2}{3} t^{3}, t\right\rangle, \quad\left(1, \frac{2}{3}, 1\right)$$

Anna Marie V.
Numerade Educator

Problem 40

Find the vectors $\mathbf{T}, \mathbf{N},$ and $\mathbf{B}$ at the given point.
$$\mathbf{r}(t)=\langle\cos t, \sin t, \ln \cos t\rangle, \quad(1,0,0)$$

Anna Marie V.
Numerade Educator

Problem 41

Find equations of the normal plane and osculating plane of the curve at the given point.
$$x=2 \sin 3 t, y=t, z=2 \cos 3 t ; \quad(0, \pi,-2)$$

Check back soon!

Problem 42

Find equations of the normal plane and osculating plane of the curve at the given point.
$$x=t, y=t^{2}, z=t^{3} ; \quad(1,1,1)$$

Check back soon!

Problem 43

Find equations of the osculating circles of the ellipse
$9 x^{2}+4 y^{2}=36$ at the points $(2,0)$ and $(0,3) .$ Use a
graphing calculator or computer to graph the ellipse and
both osculating circles on the same screen.

Check back soon!

Problem 44

Find equations of the osculating circles of the parabola
$y=\frac{1}{2} x^{2}$ at the points $(0,0)$ and $\left(1, \frac{1}{2}\right) .$ Graph both osculating circles and the parabola on the same screen.

Check back soon!

Problem 45

At what point on the curve $x=t^{3}, y=3 t, z=t^{4}$ is the
normal plane parallel to the plane $6 x+6 y-8 z=1 ?$

Check back soon!

Problem 46

Is there a point on the curve in Exercise 45 where the
osculating plane is parallel to the plane $x+y+z=1 ?$
[Note: You will need a CAS for differentiating, for sim-
plifying, and for computing a cross product.]

Check back soon!

Problem 47

Show that the curvature $\kappa$ is related to the tangent and
normal vectors by the equation
$$\frac{d \mathbf{T}}{d s}=\kappa \mathbf{N}$$

Anna Marie V.
Numerade Educator

Problem 48

Show that the curvature of a plane curve is $\kappa=|d \phi / d s|,$
where $\phi$ is the angle between $T$ and $\mathbf{i} ;$ that is, $\phi$ is the
angle of inclination of the tangent line.

Check back soon!

Problem 49

(a) Show that $d \mathbf{B} / d s$ is perpendicular to $\mathbf{B}$ .
(b) Show that $d \mathbf{B} / d s$ is perpendicular to $\mathbf{T}$ .
(c) Deduce from parts (a) and (b) that $d \mathbf{B} / d s=-\tau(s) \mathbf{N}$
for some number $\tau(s)$ called the torsion of the curve.
(The torsion measures the degree of twisting of a
curve.)
(d) Show that for a planc curve the torsion is $\tau(s)=0$ .

Anna Marie V.
Numerade Educator

Problem 50

The following formulas, called the Frenet-Serret formulas, are of fundamental importance in differential geometry:
$$\begin{array}{l}{\text { 1. } d \mathbf{T} / d s=\kappa \mathbf{N}} \\ {\text { 2. } d \mathbf{N} / d s=-\kappa \mathbf{T}+\tau \mathbf{B}} \\ {\text { 3. } d \mathbf{B} / d s=-\tau \mathbf{N}}\end{array}$$
(Formula 1 comes from Exercise 47 and Formula 3
comes from Exercise $49 .$ ) Use the fact that $\mathbf{N}=\mathbf{B} \times \mathbf{T}$
to deduce Formula 2 from Formulas 1 and 3 .

Check back soon!

Problem 51

Use the Frenct-Serret formulas to prove cach of the fol-
lowing. (Primes denote derivatives with respect to $t$ . Start
as in the proof of Theorem $10 . )$
$$\begin{array}{l}{\text { (a) } \mathbf{r}^{\prime \prime}=s^{\prime \prime} \mathbf{T}+\kappa\left(s^{\prime}\right)^{2} \mathbf{N}} \\ {\text { (b) } \mathbf{r}^{\prime} \times \mathbf{r}^{\prime \prime}=\kappa\left(s^{\prime}\right)^{3} \mathbf{B}} \\ {\text { (c) } \mathbf{r}^{\prime \prime \prime}=\left[s^{\prime \prime \prime}-\kappa^{2}\left(s^{\prime}\right)^{3}\right] \mathbf{T}+\left[3 \kappa s^{\prime} s^{\prime \prime}+\kappa^{\prime}\left(s^{\prime}\right)^{2}\right] \mathbf{N}} \\ {+\kappa \tau\left(s^{\prime}\right)^{3} \mathbf{B}}\end{array}$$
$$(d) \tau=\frac{\left(\mathbf{r}^{\prime} \times \mathbf{r}^{\prime \prime}\right) \cdot \mathbf{r}^{\prime \prime \prime}}{\left|\mathbf{r}^{\prime} \times \mathbf{r}^{\prime \prime}\right|^{2}}$$

Anna Marie V.
Numerade Educator

Problem 52

Show that the circular helix $\mathbf{r}(t)=\langle a \cos t, a \sin t, b t\rangle$
where $a$ and $b$ are positive constants, has constant curva-
ture and constant torsion. [Use the result of Exercise
51$(\mathrm{d}) . ]$

Check back soon!

Problem 53

The DNA molecule has the shape of a double helix (see
Figure 3 on page 582 ). The radius of each helix is about
10 angstroms $\left(1 \mathrm{A}=10^{-8} \mathrm{cm}\right)$ . Each helix rises about
34 A during each complete turn, and there are about
$2.9 \times 10^{8}$ complete turns. Estimate the length of each
helix.

Check back soon!

Problem 54

Let's consider the problem of designing a railroad track to
make a smooth transition between sections of straight
track. Existing track along the negative $x$ -axis is to be
joined smoothly to a track along the line $y=1$ for $x \geqslant 1$
(a) Find a polynomial $P=P(x)$ of degree 5 such that the
function $F$ defined by
$$F(x)=\left\{\begin{array}{ll}{0} & {\text { if } x \leqslant 0} \\ {P(x)} & {\text { if } 0<x<1} \\ {1} & {\text { if } x \geqslant 1}\end{array}\right.$$
is continuous and has continuous slope and continuous curvature.
(b) Usc a graphing calculator or computer to draw the
graph of $F .$

Check back soon!