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Thomas Calculus

George B. Thomas, Maurice D. Weir, Joel Hass, Frank R. Giordano

Chapter 13

Vector-Valued Functions And Motion In Space

Educators


Problem 1

In Exercises $1-4, \mathbf{r}(t)$ is the position of a particle in the $x y$ -plane at time $t .$ Find an equation in $x$ and $y$ whose graph is the path of the particle. Then find the particle s velocity and acceleration vectors at the given value of $t .$
$$
\mathbf{r}(t)=(t+1) \mathbf{i}+\left(t^{2}-1\right) \mathbf{j}, \quad t=1
$$

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Problem 2

In Exercises $1-4, \mathbf{r}(t)$ is the position of a particle in the $x y$ -plane at time $t .$ Find an equation in $x$ and $y$ whose graph is the path of the particle. Then find the particle s velocity and acceleration vectors at the given value of $t .$
$$
\mathbf{r}(t)=\left(t^{2}+1\right) \mathbf{i}+(2 t-1) \mathbf{j}, \quad t=1 / 2
$$

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Problem 3

In Exercises $1-4, \mathbf{r}(t)$ is the position of a particle in the $x y$ -plane at time $t .$ Find an equation in $x$ and $y$ whose graph is the path of the particle. Then find the particle s velocity and acceleration vectors at the given value of $t .$
$$
\mathbf{r}(t)=e^{t} \mathbf{i}+\frac{2}{9} e^{2 t} \mathbf{j}, \quad t=\ln 3
$$

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Problem 4

In Exercises $1-4, \mathbf{r}(t)$ is the position of a particle in the $x y$ -plane at time $t .$ Find an equation in $x$ and $y$ whose graph is the path of the particle. Then find the particle s velocity and acceleration vectors at the given value of $t .$
$$
\mathbf{r}(t)=(\cos 2 t) \mathbf{i}+(3 \sin 2 t) \mathbf{j}, \quad t=0
$$

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Problem 5

Exercises $5-8$ give the position vectors of particles moving along various curves in the $x y-$ plane. In each case, find the particle's velocity and acceleration vectors at the stated times and sketch them as vectors on the curve.
Motion on the circle $x^{2}+y^{2}=1$
$$
\mathbf{r}(t)=(\sin t) \mathbf{i}+(\cos t) \mathbf{j} ; \quad t=\pi / 4 \text { and } \pi / 2
$$

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Problem 6

Exercises $5-8$ give the position vectors of particles moving along various curves in the $x y-$ plane. In each case, find the particle's velocity and acceleration vectors at the stated times and sketch them as vectors on the curve.
Motion on the circle $x^{2}+y^{2}=16$
$$
\mathbf{r}(t)=\left(4 \cos \frac{t}{2}\right) \mathbf{i}+\left(4 \sin \frac{t}{2}\right) \mathbf{j} ; \quad t=\pi \text { and } 3 \pi / 2
$$

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Problem 7

Exercises $5-8$ give the position vectors of particles moving along various curves in the $x y-$ plane. In each case, find the particle's velocity and acceleration vectors at the stated times and sketch them as vectors on the curve.
Motion on the cycloid $x=t-\sin t, y=1-\cos t$
$$
\mathbf{r}(t)=\left(4 \cos \frac{t}{2}\right) \mathbf{i}+\left(4 \sin \frac{t}{2}\right) \mathbf{j} ; \quad t=\pi \text { and } 3 \pi / 2
$$

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Problem 8

Exercises $5-8$ give the position vectors of particles moving along various curves in the $x y-$ plane. In each case, find the particle's velocity and acceleration vectors at the stated times and sketch them as vectors on the curve.
Motion on the parabola $y=x^{2}+1$
$$
\mathbf{r}(t)=t \mathbf{i}+\left(t^{2}+1\right) \mathbf{j} ; \quad t=-1,0, \text { and } 1
$$

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Problem 9

In Exercises $9-14, \mathrm{r}(t)$ is the position of a particle in space at time $t .$ Find the particle's velocity and acceleration vectors. Then find the particle's speed and direction of motion at the given value of $t .$ Write the particle's velocity at that time as the product of its speed and direction.
$$
\mathbf{r}(t)=(t+1) \mathbf{i}+\left(t^{2}-1\right) \mathbf{j}+2 t \mathbf{k}, \quad t=1
$$

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Problem 10

In Exercises $9-14, \mathrm{r}(t)$ is the position of a particle in space at time $t .$ Find the particle's velocity and acceleration vectors. Then find the particle's speed and direction of motion at the given value of $t .$ Write the particle's velocity at that time as the product of its speed and direction.
$$
\mathbf{r}(t)=(1+t) \mathbf{i}+\frac{t^{2}}{\sqrt{2}} \mathbf{j}+\frac{t^{3}}{3} \mathbf{k}, \quad t=1
$$

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Problem 11

In Exercises $9-14, \mathrm{r}(t)$ is the position of a particle in space at time $t .$ Find the particle's velocity and acceleration vectors. Then find the particle's speed and direction of motion at the given value of $t .$ Write the particle's velocity at that time as the product of its speed and direction.
$$
\mathbf{r}(t)=(2 \cos t) \mathbf{i}+(3 \sin t) \mathbf{j}+4 t \mathbf{k}, \quad t=\pi / 2
$$

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Problem 12

In Exercises $9-14, \mathrm{r}(t)$ is the position of a particle in space at time $t .$ Find the particle's velocity and acceleration vectors. Then find the particle's speed and direction of motion at the given value of $t .$ Write the particle's velocity at that time as the product of its speed and direction.
$$
\mathbf{r}(t)=(\sec t) \mathbf{i}+(\tan t) \mathbf{j}+\frac{4}{3} t \mathbf{k}, \quad t=\pi / 6
$$

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Problem 13

In Exercises $9-14, \mathrm{r}(t)$ is the position of a particle in space at time $t .$ Find the particle's velocity and acceleration vectors. Then find the particle's speed and direction of motion at the given value of $t .$ Write the particle's velocity at that time as the product of its speed and direction.
$$
\mathbf{r}(t)=(2 \ln (t+1)) \mathbf{i}+t^{2} \mathbf{j}+\frac{t^{2}}{2} \mathbf{k}, \quad t=1
$$

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Problem 14

In Exercises $9-14, \mathrm{r}(t)$ is the position of a particle in space at time $t .$ Find the particle's velocity and acceleration vectors. Then find the particle's speed and direction of motion at the given value of $t .$ Write the particle's velocity at that time as the product of its speed and direction.
$$
\mathbf{r}(t)=\left(e^{-t}\right) \mathbf{i}+(2 \cos 3 t) \mathbf{j}+(2 \sin 3 t) \mathbf{k}, \quad t=0
$$

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Problem 15

In Exercises $15-18, \mathbf{r}(t)$ is the position of a particle in space at time $t .$ Find the angle between the velocity and acceleration vectors at time $t=0 .$
$$
\mathbf{r}(t)=(3 t+1) \mathbf{i}+\sqrt{3} t \mathbf{j}+t^{2} \mathbf{k}
$$

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Problem 16

In Exercises $15-18, \mathbf{r}(t)$ is the position of a particle in space at time $t .$ Find the angle between the velocity and acceleration vectors at time $t=0 .$
$$
\mathbf{r}(t)=\left(\frac{\sqrt{2}}{2} t\right) \mathbf{i}+\left(\frac{\sqrt{2}}{2} t-16 t^{2}\right) \mathbf{j}
$$

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Problem 17

In Exercises $15-18, \mathbf{r}(t)$ is the position of a particle in space at time $t .$ Find the angle between the velocity and acceleration vectors at time $t=0 .$
$$
\mathbf{r}(t)=\left(\ln \left(t^{2}+1\right)\right) \mathbf{i}+\left(\tan ^{-1} t\right) \mathbf{j}+\sqrt{t^{2}+1} \mathbf{k}
$$

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Problem 18

In Exercises $15-18, \mathbf{r}(t)$ is the position of a particle in space at time $t .$ Find the angle between the velocity and acceleration vectors at time $t=0 .$
$$
\mathbf{r}(t)=\frac{4}{9}(1+t)^{3 / 2} \mathbf{i}+\frac{4}{9}(1-t)^{3 / 2} \mathbf{j}+\frac{1}{3} t \mathbf{k}
$$

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Problem 19

In Exercises 19 and $20, \mathbf{r}(t)$ is the position vector of a particle in space at time $t .$ Find the time or times in the given time interval when the velocity and acceleration vectors are orthogonal.
$$
\mathbf{r}(t)=(t-\sin t) \mathbf{i}+(1-\cos t) \mathbf{j}, \quad 0 \leq t \leq 2 \pi
$$

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Problem 20

In Exercises 19 and $20, \mathbf{r}(t)$ is the position vector of a particle in space at time $t .$ Find the time or times in the given time interval when the velocity and acceleration vectors are orthogonal.
$$
\mathbf{r}(t)=(\sin t) \mathbf{i}+t \mathbf{j}+(\cos t) \mathbf{k}, \quad t \geq 0
$$

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Problem 21

Evaluate the integrals in Exercises $21-26$
$$
\int_{0}^{1}\left[t^{3} \mathbf{i}+7 \mathbf{j}+(t+1) \mathbf{k}\right] d t
$$

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Problem 22

Evaluate the integrals in Exercises $21-26$
$$
\int_{1}^{2}\left[(6-6 t) \mathbf{i}+3 \sqrt{t} \mathbf{j}+\left(\frac{4}{t^{2}}\right) \mathbf{k}\right] d t
$$

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Problem 23

Evaluate the integrals in Exercises $21-26$
$$
\int_{-\pi / 4}^{\pi / 4}\left[(\sin t) \mathbf{i}+(1+\cos t) \mathbf{j}+\left(\sec ^{2} t\right) \mathbf{k}\right] d t
$$

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Problem 24

Evaluate the integrals in Exercises $21-26$
$$
\int_{0}^{\pi / 3}[(\sec t \tan t) \mathbf{i}+(\tan t) \mathbf{j}+(2 \sin t \cos t) \mathbf{k}] d t
$$

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Problem 25

Evaluate the integrals in Exercises $21-26$
$$
\int_{1}^{4}\left[\frac{1}{t} \mathbf{i}+\frac{1}{5-t} \mathbf{j}+\frac{1}{2 t} \mathbf{k}\right] d t
$$

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Problem 26

Evaluate the integrals in Exercises $21-26$
$$
\int_{0}^{1}\left[\frac{2}{\sqrt{1-t^{2}}} \mathbf{i}+\frac{\sqrt{3}}{1+t^{2}} \mathbf{k}\right] d t
$$

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Problem 27

Solve the initial value problems in Exercises $27-32$ for $r$ as a vector
function of $t .$
$$
\begin{array}{ll}{\text { Differential equation: }} & {\frac{d \mathbf{r}}{d t}=-t \mathbf{i}-t \mathbf{j}-t \mathbf{k}} \\ {\text { Initial condition: }} & {\mathbf{r}(0)=\mathbf{i}+2 \mathbf{j}+3 \mathbf{k}}\end{array}
$$

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Problem 28

Solve the initial value problems in Exercises $27-32$ for $r$ as a vector
function of $t .$
$$
\begin{array}{ll}{\text { Differential equation: }} & {\frac{d \mathbf{r}}{d t}=(180 t) \mathbf{i}+\left(180 t-16 t^{2}\right) \mathbf{j}} \\ {\text { Initial condition: }} & {\mathbf{r}(0)=100 \mathbf{j}}\end{array}
$$

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Problem 29

Solve the initial value problems in Exercises $27-32$ for $r$ as a vector
function of $t .$
$$
\begin{array}{ll}{\text { Differential equation: }} & {\frac{d \mathbf{r}}{d t}=\frac{3}{2}(t+1)^{1 / 2} \mathbf{i}+e^{-t} \mathbf{j}+\frac{1}{t+1} \mathbf{k}} \\ {\text { Initial condition: }} & {\mathbf{r}(0)=\mathbf{k}}\end{array}
$$

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Problem 30

Solve the initial value problems in Exercises $27-32$ for $r$ as a vector
function of $t .$
$$
\begin{array}{ll}{\text { Differential equation: }} & {\frac{d \mathbf{r}}{d t}=\left(t^{3}+4 t\right) \mathbf{i}+t \mathbf{j}+2 t^{2} \mathbf{k}} \\ {\text { Initial condition: }} & {\mathbf{r}(0)=\mathbf{i}+\mathbf{j}}\end{array}
$$

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Problem 31

Solve the initial value problems in Exercises $27-32$ for $r$ as a vector
function of $t .$
$$
\begin{array}{ll}{\text { Differential equation: }} & {\frac{d^{2} \mathbf{r}}{d t^{2}}=-32 \mathbf{k}} \\ {\text { Initial conditions: }} & {\mathbf{r}(0)=100 \mathbf{k} \text { and }} \\ {} & {\left.\frac{d \mathbf{r}}{d t}\right|_{t=0}=8 \mathbf{i}+8 \mathbf{j}}\end{array}
$$

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Problem 32

Solve the initial value problems in Exercises $27-32$ for $r$ as a vector
function of $t .$
$$
\begin{array}{ll}{\text { Differential equation: }} & {\frac{d^{2} \mathbf{r}}{d t^{2}}=-(\mathbf{i}+\mathbf{j}+\mathbf{k})} \\ {\text { Initial conditions: }} & {\frac{d^{2} \mathbf{r}}{d t^{2}}=10 \mathbf{i}+10 \mathbf{j}+10 \mathbf{k} \text { and }} \\ {} & {\left.\frac{d \mathbf{r}}{d t}\right|_{t=0}=0}\end{array}
$$

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Problem 33

As mentioned in the text, the tangent line to a smooth curve $\mathbf{r}(t)=f(t) \mathbf{i}+g(t) \mathbf{j}+h(t) \mathbf{k}$ at $t=t_{0}$ is the line that passes through the point $\left(f\left(t_{0}\right), g\left(t_{0}\right), h\left(t_{0}\right)\right)$ parallel to $\mathbf{v}\left(t_{0}\right),$ the curve's velocity vector at $t_{0} .$ In Exercises $33-36,$ find parametric equations for the line that is tangent to the given curve at the given parameter value $t=t_{0}$ .
$$
\mathbf{r}(t)=(\sin t) \mathbf{i}+\left(t^{2}-\cos t\right) \mathbf{j}+e^{t} \mathbf{k}, \quad t_{0}=0
$$

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Problem 34

As mentioned in the text, the tangent line to a smooth curve $\mathbf{r}(t)=f(t) \mathbf{i}+g(t) \mathbf{j}+h(t) \mathbf{k}$ at $t=t_{0}$ is the line that passes through the point $\left(f\left(t_{0}\right), g\left(t_{0}\right), h\left(t_{0}\right)\right)$ parallel to $\mathbf{v}\left(t_{0}\right),$ the curve's velocity vector at $t_{0} .$ In Exercises $33-36,$ find parametric equations for the line that is tangent to the given curve at the given parameter value $t=t_{0}$ .
$$
\mathbf{r}(t)=(2 \sin t) \mathbf{i}+(2 \cos t) \mathbf{j}+5 t, \quad t_{0}=4 \pi
$$

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Problem 35

As mentioned in the text, the tangent line to a smooth curve $\mathbf{r}(t)=f(t) \mathbf{i}+g(t) \mathbf{j}+h(t) \mathbf{k}$ at $t=t_{0}$ is the line that passes through the point $\left(f\left(t_{0}\right), g\left(t_{0}\right), h\left(t_{0}\right)\right)$ parallel to $\mathbf{v}\left(t_{0}\right),$ the curve's velocity vector at $t_{0} .$ In Exercises $33-36,$ find parametric equations for the line that is tangent to the given curve at the given parameter value $t=t_{0}$ .
$$
\mathbf{r}(t)=(a \sin t) \mathbf{i}+(a \cos t) \mathbf{j}+b t \mathbf{k}, \quad t_{0}=2 \pi
$$

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Problem 36

As mentioned in the text, the tangent line to a smooth curve $\mathbf{r}(t)=f(t) \mathbf{i}+g(t) \mathbf{j}+h(t) \mathbf{k}$ at $t=t_{0}$ is the line that passes through the point $\left(f\left(t_{0}\right), g\left(t_{0}\right), h\left(t_{0}\right)\right)$ parallel to $\mathbf{v}\left(t_{0}\right),$ the curve's velocity vector at $t_{0} .$ In Exercises $33-36,$ find parametric equations for the line that is tangent to the given curve at the given parameter value $t=t_{0}$ .
$$
\mathbf{r}(t)=(\cos t) \mathbf{i}+(\sin t) \mathbf{j}+(\sin 2 t) \mathbf{k}, \quad t_{0}=\frac{\pi}{2}
$$

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Problem 37

Each of the following equations in parts (a)-(e) describes the motion of a particle having the same path, namely the unit circle $x^{2}+y^{2}=1$ . Although the path of each particle in parts (a)- (e) is the same, the behavior, or "dynamics," of each particle is different. For each particle, answer the following questions.
i. Does the particle have constant speed? If so, what is its constant speed?
ii. Is the particle's acceleration vector always orthogonal to its velocity vector?
iii. Does the particle move clockwise or counterclockwise around the circle?
$$
\begin{array}{l}{\text { a. } \mathbf{r}(t)=(\cos t) \mathbf{i}+(\sin t) \mathbf{j}, \quad t \geq 0} \\ {\text { b. } \mathbf{r}(t)=\cos (2 t) \mathbf{i}+\sin (2 t) \mathbf{j}, \quad t \geq 0} \\ {\text { c. } \mathbf{r}(t)=\cos (t-\pi / 2) \mathbf{i}+\sin (t-\pi / 2) \mathbf{j}, \quad t \geq 0} \\ {\text { d. } \mathbf{r}(t)=(\cos t) \mathbf{i}-(\sin t) \mathbf{j}, \quad t \geq 0} \\ {\text { e. } \mathbf{r}(t)=\cos \left(t^{2}\right) \mathbf{i}+\sin \left(t^{2}\right) \mathbf{j}, \quad t \geq 0}\end{array}
$$

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Problem 38

Show that the vector-valued function
$\mathbf{r}(t)=(2 \mathbf{i}+2 \mathbf{j}+\mathbf{k})$
$$
+\cos t\left(\frac{1}{\sqrt{2}} \mathbf{i}-\frac{1}{\sqrt{2}} \mathbf{j}\right)+\sin t\left(\frac{1}{\sqrt{3}} \mathbf{i}+\frac{1}{\sqrt{3}} \mathbf{j}+\frac{1}{\sqrt{3}} \mathbf{k}\right)
$$
describes the motion of a particle moving in the circle of radius 1 centered at the point $(2,2,1)$ and lying in the plane $x+y-2 z=2$ .

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Problem 39

At time $t=0,$ a particle is located at the point $(1,2,3) .$ It travels in a straight line to the point $(4,1,4),$ has speed 2 at $(1,2,3)$ and constant acceleration $3 \mathbf{i}-\mathbf{j}+\mathbf{k}$ . Find an equation for the position vector $\mathbf{r}(t)$ of the particle at time $t .$

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Problem 40

A particle traveling in a straight line is located at the point $(1,-1,2)$ and has speed 2 at time $t=0$ . The particle moves toward the point $(3,0,3)$ with constant acceleration $2 \mathbf{i}+\mathbf{j}+\mathbf{k}$ . Find its position vector $\mathbf{r}(t)$ at time $t .$

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Problem 41

Motion along a parabola A particle moves along the top of the parabola $y^{2}=2 x$ from left to right at a constant speed of 5 units per second. Find the velocity of the particle as it moves through the point $(2,2) .$

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Problem 42

Motion along a cycloid A particle moves in the $x y$ -plane in such a way that its position at time $t$ is
$$
\mathbf{r}(t)=(t-\sin t) \mathbf{i}+(1-\cos t) \mathbf{j}
$$
a. Graph $\mathbf{r}(t) .$ The resulting curve is a cycloid.
b. Find the maximum and minimum values of $|\mathbf{v}|$ and $|\mathbf{a}| .$ (Hint: Find the extreme values of $|\mathbf{v}|^{2}$ and $|\mathbf{a}|^{2}$ first and take square roots later.)

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Problem 43

Motion along an ellipse A particle moves around the ellipse
$(y / 3)^{2}+(z / 2)^{2}=1$ in the $y z$ -plane in such a way that its position at time $t$ is
$$
\mathbf{r}(t)=(3 \cos t) \mathbf{j}+(2 \sin t) \mathbf{k}
$$
Find the maximum and minimum values of $|\mathbf{v}|$ and $|\mathbf{a}| .$ (Hint: Find the extreme values of $|\mathbf{v}|^{2}$ and $|\mathbf{a}|^{2}$ first and take square roots later.)

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Problem 44

A satellite in circular orbit $A$ satellite of mass $m$ is revolving at a constant speed $v$ around a body of mass $M$ (Earth, for example $)$ in a circular orbit of radius $r_{0}$ (measured from the body's center of mass). Determine the satellite's orbital period $T$ (the time to complete one full orbit), as follows:

a. Coordinatize the orbital plane by placing the origin at the body's center of mass, with the satellite on the $x$ -axis at $t=0$ and moving counterclockwise, as in the accompanying figure.
Let $\mathbf{r}(t)$ be the satellite's position vector at time $t .$ Show that $\theta=y t / r_{0}$ and hence that
$$
r(t)=\left(r_{0} \cos \frac{v t}{r_{0}}\right) \mathbf{i}+\left(r_{0} \sin \frac{v t}{r_{0}}\right) \mathbf{j}
$$
b. Find the acceleration of the satellite.
c. According to Newton's law of gravitation, the gravitational force exerted on the satellite is directed toward $M$ and is given by
$$
\mathbf{F}=\left(-\frac{G m M}{r_{0}^{2}}\right) \frac{\mathbf{r}}{r_{0}}
$$
where $G$ is the universal constant of gravitation. Using Newton's second law, $\mathbf{F}=m \mathbf{a},$ show that $v^{2}=G M / r_{0}$ .
d. Show that the orbital period $T$ satisfies $v T=2 \pi r_{0}$ .
e. From parts (c) and (d), deduce that
$$
T^{2}=\frac{4 \pi^{2}}{G M} r_{0}^{3}
$$
That is, the square of the period of a satellite in circular orbit is proportional to the cube of the radius from the orbital center.

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Problem 45

Let $v$ be a differentiable vector function of $t .$ Show that if $v \cdot(d v / d t)=0$ for all $t,$ then $|v|$ is constant.

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Problem 46

Derivatives of triple scalar products
a. Show that if $\mathbf{u}, \mathbf{v},$ and $\mathbf{w}$ are differentiable vector functor functons of
$t,$ then
$$
\begin{array}{r}{\frac{d}{d t}(\mathbf{u} \cdot \mathbf{v} \times \mathbf{w})=\frac{d \mathbf{u}}{d t} \cdot \mathbf{v} \times \mathbf{w}+\mathbf{u} \cdot \frac{d \mathbf{v}}{d t} \times \mathbf{w}+} \\ {\mathbf{u} \cdot \mathbf{v} \times \frac{d \mathbf{w}}{d t}}\end{array}
$$
b. Show that Equation $(7)$ is equivalent to
$$
\frac{d}{d t}\left|\begin{array}{ccc}{u_{1}} & {u_{2}} & {u_{3}} \\ {v_{1}} & {v_{2}} & {v_{3}} \\ {w_{1}} & {w_{2}} & {w_{3}}\end{array}\right|=\left|\begin{array}{ccc}{\frac{d u_{1}}{d t}} & {\frac{d u_{2}}{d t}} & {\frac{d u_{3}}{d t}} \\ {v_{1}} & {v_{2}} & {v_{3}} \\ {w_{1}} & {w_{2}} & {w_{3}}\end{array}\right|
$$
$$
+\left|\begin{array}{lll}{u_{1}} & {u_{2}} & {u_{3}} \\ {\frac{d v_{1}}{d t}} & {\frac{d v_{2}}{d t}} & {\frac{d v_{3}}{d t}} \\ {w_{1}} & {w_{2}} & {w_{3}}\end{array}\right|
$$
$$
+\left|\begin{array}{lll}{u_{1}} & {u_{2}} & {u_{3}} \\ {v_{1}} & {v_{2}} & {v_{3}} \\ {\frac{d w_{1}}{d t}} & {\frac{d w_{2}}{d t}} & {\frac{d w_{3}}{d t}}\end{array}\right|
$$

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Problem 47

(Continuation of Exercise $46 .$ ) Suppose that $\mathbf{r}(t)=f(t) \mathbf{i}+$ $g(t) \mathbf{j}+h(t) \mathbf{k}$ and that $f, g,$ and $h$ have derivatives through order three. Use Equation $(7)$ or $(8)$ to show that \
$$
\frac{d}{d t}\left(\mathbf{r} \cdot \frac{d \mathbf{r}}{d t} \times \frac{d^{2} \mathbf{r}}{d t^{2}}\right)=\mathbf{r} \cdot\left(\frac{d \mathbf{r}}{d t} \times \frac{d^{3} \mathbf{r}}{d t^{3}}\right)
$$
(Hint: Differentiate on the left and look for vectors whose products are zero.)

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Problem 48

Constant Function Rule Prove that if $\mathbf{u}$ is the vector function with the constant value $\mathbf{C},$ then $d \mathbf{u} / d t=\mathbf{0} .$

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Problem 49

Scalar Multiple Rules
a. Prove that if $\mathbf{u}$ is a differentiable function of $t$ and $c$ is any real number, then
$$
\frac{d(c \mathbf{u})}{d t}=c \frac{d \mathbf{u}}{d t}
$$
b. Prove that if $\mathbf{u}$ is a differentiable function of $t$ and $f$ is a differentiable scalar function of $t,$ then
$$
\frac{d}{d t}(f \mathbf{u})=\frac{d f}{d t} \mathbf{u}+f \frac{d \mathbf{u}}{d t}
$$

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Problem 50

Sum and Difference Rules Prove that if $\mathbf{u}$ and $\mathbf{v}$ are differentiable functions of $t,$ then
$$
\frac{d}{d t}(\mathbf{u}+\mathbf{v})=\frac{d \mathbf{u}}{d t}+\frac{d \mathbf{v}}{d t}
$$
and
$$
\frac{d}{d t}(\mathbf{u}-\mathbf{v})=\frac{d \mathbf{u}}{d t}-\frac{d \mathbf{v}}{d t}
$$

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Problem 51

Component Test for Continuity at a Point Show that the vector function $\mathbf{r}$ defined by $\mathbf{r}(t)=f(t) \mathbf{i}+g(t) \mathbf{j}+h(t) \mathbf{k}$ is continuous at $t=t_{0}$ if and only if $f, g,$ and $h$ are continuous at $t_{0} .$

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Problem 52

Limits of cross products of vector functions Suppose that $\mathbf{r}_{1}(t)=f_{1}(t) \mathbf{i}+f_{2}(t) \mathbf{j}+f_{3}(t) \mathbf{k}, \mathbf{r}_{2}(t)=g_{1}(t) \mathbf{i}+g_{2}(t) \mathbf{j}+$
$g_{3}(t) \mathbf{k}, \lim _{t \rightarrow t_{0}} \mathbf{r}_{1}(t)=\mathbf{A},$ and $\lim _{t \rightarrow t_{0}} \mathbf{r}_{2}(t)=\mathbf{B}$ . Use the minant formula for cross products and the Limit Product Rule for scalar functions to show that
$$
\lim _{t \rightarrow t_{0}}\left(\mathbf{r}_{1}(t) \times \mathbf{r}_{2}(t)\right)=\mathbf{A} \times \mathbf{B}
$$

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Problem 53

Differentiable vector functions are continuous Show that if $\mathbf{r}(t)=f(t) \mathbf{i}+g(t) \mathbf{j}+h(t) \mathbf{k}$ is differentiable at $t=t_{0},$ then it is continuous at $t_{0}$ as well.

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Problem 54

Establish the following properties of integrable vector functions.
a. The Constant Scalar Multiple Rule:
$$
\int_{a}^{b} k \mathbf{r}(t) d t=k \int_{a}^{b} \mathbf{r}(t) d t \quad \text { (any scalar } k )
$$
The Rule for Negatives,
$$
\int_{a}^{b}(-\mathbf{r}(t)) d t=-\int_{a}^{b} \mathbf{r}(t) d t
$$
$\quad$ is obtained by taking $k=-1$
b. The Sum and Difference Rules:
$$
\int_{a}^{b}\left(\mathbf{r}_{1}(t) \pm \mathbf{r}_{2}(t)\right) d t=\int_{a}^{b} \mathbf{r}_{1}(t) d t \pm \int_{a}^{b} \mathbf{r}_{2}(t) d t
$$
c. The Constant Vector Multiple Rules:
$$
\begin{array}{l}{\int_{a}^{b} \mathbf{C} \cdot \mathbf{r}(t) d t=\mathbf{C} \cdot \int_{a}^{b} \mathbf{r}(t) d t \quad \text { (any constant vector } \mathbf{C} )} \\ {\text { and }} \\ {\int_{a}^{b} \mathbf{C} \times \mathbf{r}(t) d t=\mathbf{C} \times \int_{a}^{b} \mathbf{r}(t) d t \quad \text { (any constant vector } \mathbf{C} )}\end{array}
$$

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Problem 55

Products of scalar and vector functions Suppose that the scalar function $u(t)$ and the vector function $\mathbf{r}(t)$ are both defined for $a \leq t \leq b$ .
a. Show that $u \mathbf{r}$ is continuous on $[a, b]$ if $u$ and $\mathbf{r}$ are continuous
on $[a, b] .$ If $u$ and $\mathbf{r}$ are both differentiable on $[a, b],$ show that $u \mathbf{r}$ is
differentiable on $[a, b]$ and that
$$
\frac{d}{d t}(u \mathbf{r})=u \frac{d \mathbf{r}}{d t}+\mathbf{r} \frac{d u}{d t}
$$

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Problem 56

Antiderivatives of vector functions
a. Use Corollary 2 of the Mean Value Theorem for scalar functions to show that if two vector functions $\mathbf{R}_{1}(t)$ and $\mathbf{R}_{2}(t)$ have identical derivatives on an interval $I$ , then the functions differ by a constant vector value throughout $I .$
b. Use the result in part (a) to show that if $\mathbf{R}(t)$ is any anti- derivative of $\mathbf{r}(t)$ on $I,$ then any other antiderivative of $\mathbf{r}$ on $I$ equals $\mathbf{R}(t)+\mathbf{C}$ for some constant vector $\mathbf{C}$ .

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Problem 57

The Fundamental Theorem of Calculus The Fundamental Theorem of Calculus for scalar functions of a real variable holds for vector functions of a real variable as well. Prove this by using he theorem for scalar functions to show first that if a vector funcion $\mathbf{r}(t)$ is continuous for $a \leq t \leq b,$ then
$$
\frac{d}{d t} \int_{a}^{t} \mathbf{r}(\tau) d \tau=\mathbf{r}(t)
$$
at every point $t$ of $(a, b) .$ Then use the conclusion in part (b) of Exercise 56 to show that if $R$ is any antiderivative of $r$ on $[a, b]$ then
$$
\int_{a}^{b} \mathbf{r}(t) d t=\mathbf{R}(b)-\mathbf{R}(a)
$$

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Problem 58

Use a CAS to perform the following steps in Exercises $58-61 .$
a. Plot the space curve traced out by the position vector $\mathbf{r}$ .
b. Find the components of the velocity vector $d \mathbf{r} / d t$ .
c. Evaluate $d \mathbf{r} / d t$ at the given point $t_{0}$ and determine the equation of the tangent line to the curve at $\mathbf{r}\left(t_{0}\right) .$
d. Plot the tangent line together with the curve over the given interval.
$$
\begin{array}{l}{\mathbf{r}(t)=(\sin t-t \cos t) \mathbf{i}+(\cos t+t \sin t) \mathbf{j}+t^{2} \mathbf{k}} \\ {0 \leq t \leq 6 \pi, \quad t_{0}=3 \pi / 2}\end{array}
$$

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Problem 59

Use a CAS to perform the following steps in Exercises $58-61 .$
a. Plot the space curve traced out by the position vector $\mathbf{r}$ .
b. Find the components of the velocity vector $d \mathbf{r} / d t$ .
c. Evaluate $d \mathbf{r} / d t$ at the given point $t_{0}$ and determine the equation of the tangent line to the curve at $\mathbf{r}\left(t_{0}\right) .$
d. Plot the tangent line together with the curve over the given interval.
$$
\mathbf{r}(t)=\sqrt{2 t \mathbf{i}}+e^{t} \mathbf{j}+e^{-t} \mathbf{k}, \quad-2 \leq t \leq 3, \quad t_{0}=1
$$

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Problem 60

Use a CAS to perform the following steps in Exercises $58-61 .$
a. Plot the space curve traced out by the position vector $\mathbf{r}$ .
b. Find the components of the velocity vector $d \mathbf{r} / d t$ .
c. Evaluate $d \mathbf{r} / d t$ at the given point $t_{0}$ and determine the equation of the tangent line to the curve at $\mathbf{r}\left(t_{0}\right) .$
d. Plot the tangent line together with the curve over the given interval.
$$
\begin{array}{l}{\mathbf{r}(t)=(\sin 2 t) \mathbf{i}+(\ln (1+t)) \mathbf{j}+t \mathbf{k}, \quad 0 \leq t \leq 4 \pi} \\ {t_{0}=\pi / 4}\end{array}
$$

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Problem 61

Use a CAS to perform the following steps in Exercises $58-61 .$
a. Plot the space curve traced out by the position vector $\mathbf{r}$ .
b. Find the components of the velocity vector $d \mathbf{r} / d t$ .
c. Evaluate $d \mathbf{r} / d t$ at the given point $t_{0}$ and determine the equation of the tangent line to the curve at $\mathbf{r}\left(t_{0}\right) .$
d. Plot the tangent line together with the curve over the given interval.
$$
\begin{array}{l}{\mathbf{r}(t)=\left(\ln \left(t^{2}+2\right)\right) \mathbf{i}+\left(\tan ^{-1} 3 t\right) \mathbf{j}+\sqrt{t^{2}+1} \mathbf{k}} \\ {-3 \leq t \leq 5, \quad t_{0}=3}\end{array}
$$

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Problem 62

In Exercises 62 and $63,$ you will explore graphically the behavior of the helix
$$
\mathbf{r}(t)=(\cos a t) \mathbf{i}+(\sin a t) \mathbf{j}+b t \mathbf{k}
$$
as you change the values of the constants $a$ and $b$ . Use a CAS to perform the steps in each exercise.

Set $b=1 .$ Plot the helix $r(t)$ together with the tangent line to the curve at $t=3 \pi / 2$ for $a=1,2,4,$ and 6 over the interval $0 \leq t \leq 4 \pi$ . Describe in your own words what happens to the graph of the helix and the position of the tangent line as $a$ increases through these positive values.

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Problem 63

In Exercises 62 and $63,$ you will explore graphically the behavior of the helix
$$
\mathbf{r}(t)=(\cos a t) \mathbf{i}+(\sin a t) \mathbf{j}+b t \mathbf{k}
$$
as you change the values of the constants $a$ and $b$ . Use a CAS to perform the steps in each exercise.

Set $a=1 .$ Plot the helix $\mathbf{r}(t)$ together with the tangent line to the curve at $t=3 \pi / 2$ for $b=1 / 4,1 / 2,2,$ and 4 over the interval $0 \leq t \leq 4 \pi$ . Describe in your own words what happens to the graph of the helix and the position of the tangent line as $b$ increases through these positive values.

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