Deck 10: Vector Functions

Question 1

Find a parametric representation for the surface consisting of that part of the cylinder that lies between the planes and y = 3.

Answer 1

, , , ,

Question 2

Are the two planes $\mathbf { r } _ { 1 } ( s , t ) = \langle 1 + s + t , s - t , 1 + 2 s \rangle \text { and } \mathbf { r } _ { 2 } ( s , t ) = \langle 2 + s + 2 t , 3 + t , s + 3 t \rangle$ parallel? Justify your answer.

Answer 2

True

Question 3

Find a parametric representation for the surface $z = x ^ { 2 } + y ^ { 2 }$

A) $x = r \sin \theta$ , $y = r \sin \theta$ , $z = r$

B) $x = r \sin \theta$ , $y = r \sin \theta$ , $z = r ^ { 2 }$
C) $x = r \cos \theta$ , $y = r \cos \theta$ , $z = r$
D) $x = r \cos \theta$ , $y = r \cos \theta$ , $z = r ^ { 2 }$
E) $x = \cos \theta$ , $y = \sin \theta$ , $z = r$
F) $x = \cos \theta$ , $y = \sin \theta$ , $z = r ^ { 2 }$

G) $x = r \cos \theta$ , $y = r \sin \theta$ , $z = r$

H) $x = r \cos \theta$ , $y = r \sin \theta$ , $z = r ^ { 2 }$

Answer 3

$x = r \cos \theta$ , $y = r \sin \theta$ , $z = r ^ { 2 }$

Question 4

A picture of a circular cylinder with radius a and height h is given below. Find a parametric representation of the cylinder.

Question 5

Identify the geometric object that is represented by parametric equations $\mathbf { r } ( t , s ) = \langle 3 \cos s , 3 \sin s , t \rangle$ .

A)A plane
B)A cone
C)A straight line
D)A circular cylinder
E)A circle
F)A circular disk
G)A helix
H)A sphere

Question 6

Find a parametric representation for the surface consisting of that part of the hyperboloid .

Question 7

Identify the geometric object that is represented by parametric equations $\mathbf { r } ( t ) = \{ 3 \cos t , 3 \sin t , 5 \}$ .

A)A plane
B)A cone
C)A straight line
D)A circular cylinder
E)A circle
F)A circular disk
G)A helix
H)A sphere

Question 8

Identify the geometric object that is represented by parametric equations $\mathbf { r } ( t , s ) = \langle t \cos s , t \sin s , t \rangle$ .

A)A plane
B)A cone
C)A straight line
D)A circular cylinder
E)A circle
F)A circular disk
G)A helix
H)A sphere

Question 9

Find a parametric representation for the surface consisting of that part of the hyperboloid that lies below the rectangle .

Question 10

Identify the surface with the vector equation . (Hint: First consider .)

Question 11

Identify the geometric object that is represented by parametric equations $\mathbf { r } ( t ) = \langle 3 \cos t , 3 \sin t , t \rangle$ .

A)A plane
B)A cone
C)A straight line
D)A circular cylinder
E)A circle
F)A circular disk
G)A helix
H)A sphere

Question 12

Identify the geometric object that is represented by parametric equations $\mathbf { r } ( t , s ) = \langle s + t , 3 t + 1,3 s - 5 t \rangle$ .

A)A plane
B)A cone
C)A straight line
D)A circular cylinder
E)A circle
F)A circular disk
G)A helix
H)A sphere

Question 13

Find a parametric representation for the surface consisting of that part of the elliptic paraboloid that lies in front of the plane x = 0.

Question 14

Identify the geometric object that is represented by parametric equations $\mathbf { r } ( t ) = \langle 1 + t , 3 t , 3 - 5 t \rangle$ .

A)A plane
B)A cone
C)A straight line
D)A circular cylinder
E)A circle
F)A circular disk
G)A helix
H)A sphere

Question 15

Let the position function of a particle be $\mathbf { r } ( t ) = \left\langle t ^ { 2 } , 2 t , e ^ { t } \right\rangle$ . Find the velocity of the particle when t = 1.

A) $( 2,2,1 )$

B) $( 2,2 , e )$
C) $(2,0,1 )$
D) $( 2,0 , e )$
E) $( 1,1,1 )$
F) $( 1,1 , e )$

G) $( 1,0,1 )$

H) $( 1,0 , e )$

Question 16

Identify the geometric object that is represented by parametric equations $\mathbf { r } ( t , s ) = \langle 3 \sin s \cos t , 3 \sin s \sin t , 3 \cos s \rangle$ .

A)A plane
B)A cone
C)A straight line
D)A circular cylinder
E)A circle
F)A circular disk
G)A helix
H)A sphere

Question 17

Find a parametric representation for the surface consisting of the upper half of the ellipsoid $x ^ { 2 } + 5 y ^ { 2 } + z ^ { 2 } = 1$ .

A) $x = x , y = y , z = \sqrt { 1 + x ^ { 2 } + 5 y ^ { 2 } }$

B) $x = x , y = y , z = \sqrt { 1 + x ^ { 2 } - 5 y ^ { 2 } }$
C) $x = x , y = y , z = \sqrt { 1 - x ^ { 2 } + 5 y ^ { 2 } }$
D) $x = x , y = y , z = \sqrt { 1 - x ^ { 2 } - 5 y ^ { 2 } }$
E) $x = x , y = y , z = \sqrt { x ^ { 2 } + 5 y ^ { 2 } - 1 }$
F) $x = x , y = y , z = \sqrt { x ^ { 2 } - 5 y ^ { 2 } - 1 }$

G) $x = x , y = y , z = \sqrt { - x ^ { 2 } + 5 y ^ { 2 } - 1 }$

H) $x = x , y = y , z = \sqrt { - x ^ { 2 } - 5 y ^ { 2 } - 1 }$

Question 18

Identify the surface with the vector equation

Question 19

Identify the geometric object that is represented by parametric equations $\mathbf { r } ( t , s ) = \langle s \cos t , s \sin t , t \rangle$ .

A)A plane
B)A cone
C)A straight line
D)A circular cylinder
E)A circle
F)A circular disk
G)A helix
H)A sphere

Question 20

Find a parametric representation for the surface consisting of that part of the plane z = x + 3 that lies inside the cylinder .

Question 21

Let the position function of a particle be $\mathbf { r } ( t ) = t \mathbf { i } + t ^ { 2 } \mathbf { j }$ . Find the tangential component of the acceleration vector when t = 1.

A) $\frac { 1 } { \sqrt { 5 } }$

B) $\frac { 2 } { \sqrt { 5 } }$
C) $\frac { 3 } { \sqrt { 5 } }$
D) $\frac { 4 } { \sqrt { 5 } }$
E) $\sqrt { 5 }$
F) $\frac { 6 } { \sqrt { 5 } }$

G) $\frac { 7 } { \sqrt { 5 } }$

H) $\frac { 8 } { \sqrt { 5 } }$

Question 22

Let the velocity of a particle be $\mathbf { v } ( t ) = \mathbf { i } + t \mathbf { j }$ , and let its position when t = 0 be $\mathbf { r } ( 0 ) = \mathbf { j } + 2 \mathbf { k }$ . Find its position when t = 2.

A) $3 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }$

B) $2 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }$
C) $3 \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k }$
D) $3 \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k }$
E) $3 \mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k }$
F) $2 \mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k }$

G) $4 \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k }$

H) $4 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }$

Question 23

Suppose a particle moves in the plane according to the vector-valued function , where t represents time. Find , and sketch a graph showing the path taken by the particle indicating the direction of motion.

Question 24

For , find and , the tangential and normal components of acceleration.

Question 25

Let the position function of a particle be $\mathbf { r } ( t ) = \left\langle t ^ { 2 } , 2 t , e ^ { t } \right\rangle$ . Find the acceleration of the particle when t = 0.

A) $( 2,2,1 )$

B) $( 2,2 , e )$
C) $( 2,0,1 )$
D) $( 2,0 , e )$
E) $(1,1,1 )$
F) $( 1,1 , e )$

G) $( 1,0,1 )$

H) $( 1,0 , e )$

Question 26

A particle is traveling along a helix whose vector equation is given by . Show that its velocity and acceleration are orthogonal at all times.

Question 27

A paper carrier is traveling 60 miles per hour down a straight road in the direction of the vector i when he throws a paper out the car window with a velocity (relative to the car) in the direction of j and of magnitude 10 miles per hour.(a) Find the velocity of the paper relative to the ground when the paper carrier releases it.(b) Find the speed of the paper at that time.

Question 28

Let a (t), v (t), and r (t) denote the acceleration, velocity, and position at time t of an object moving in the xy-plane. Find r (t), given that

Question 29

Let the position function of a particle be $\mathbf { r } ( t ) = \langle 3 \sin 2 t , 2 \cos 2 t , - \sin 4 t \rangle$ . Find the speed of the particle when $t = \frac { \pi } { 4 }$ .

A)1
B)4
C)3
D) $4 \sqrt { 2 }$
E) $\sqrt { 8 }$
F) $\sqrt { 10 }$
G) $\sqrt { 13 }$
H) $\sqrt { 14 }$

Question 30

A person is standing 80 feet from a tall cliff. She throws a rock at 80 feet per second at an angle of 45° from the horizontal. Neglecting air resistance and discounting the height of the person, how far up the cliff does it hit?

Question 31

Is it possible for the velocity of a particle to be zero at the same time its acceleration is not zero? Explain.

Question 32

Let the position function of a particle be $\mathbf { r } ( t ) = \left\langle t ^ { 2 } , 1 - 2 t , t \right\rangle$ . Find the smallest value of its speed.

A)0
B)1
C) $\sqrt { 2 }$
D) $\sqrt { 3 }$
E)2
F) $\sqrt { 5 }$
G) $\sqrt { 6 }$
H) $\sqrt { 7 }$

Question 33

Let the acceleration of a particle be $\mathbf { a } ( t ) = t \mathbf { i }$ , and let its velocity when t = 0 be $\mathbf { v } ( 0 ) = \mathbf { i } + \mathbf { k }$ . Find its speed when t = 2.

A) $\sqrt { 5 }$
B) $\sqrt { 6 }$
C) $\sqrt { 7 }$
D) $\sqrt { 8 }$
E)3
F) $\sqrt { 10 }$
G) $\sqrt { 11 }$
H) $\sqrt { 12 }$

Question 34

Let the acceleration of a particle be $\mathbf { a } ( t ) = \mathbf { i } + \mathbf { k }$ , and let its velocity when t = 0 be $\mathbf { v } ( 0 ) = \mathbf { j }$ . Find its velocity when t = 1.

A) $\frac { 1 } { 2 } \mathbf { i } + \mathbf { k }$
B) $2 \mathbf { i } + \mathbf { k }$
C) $3 \mathbf { i } + \mathbf { k }$
D) $\mathbf { i } + 2 \mathbf { k }$
E) $\frac { 1 } { 2 } \mathbf { i }$
F) $\mathbf { i } + \mathbf { k }$
G) $\mathbf { i } + \mathbf { j } + \mathbf { k }$
H) $\mathbf { j } + \mathbf { k }$

Question 35

Suppose a particle is moving in the xy-plane so that its position vector at time t is given by . Find the velocity, speed, and acceleration of the particle at time t = 2.

Question 36

If a particle moves in a plane with constant acceleration, show that its path is a straight line or a parabola.

Question 37

Floyd Thunderfoot is a punter for the Vikings. Today the Vikings are playing the Bears in the Metrodome. The Bears stop the Vikings at the Vikings' 40 yard line (line of scrimmage), and Floyd is called in to punt. Floyd needs to kick from 10 yards behind the line of scrimmage in order to get the punt off in time. If the ball has a hang time of 4 seconds and lands at the Bears' 10 yard line, at what angle did Floyd kick the ball, and at what speed? (Ignore air resistance.)

Question 38

Let the position function of a particle be r (t) = sin 3t i+cos 3t j+sin 4t k. Find the smallest value of its speed.

A)1
B)2
C)9
D) $\sqrt { 3 }$
E)0
F) $\sqrt { 10 }$
G) $\sqrt { 2 }$
H)3

Question 39

Let the position function of a particle be $\mathbf { r } ( t ) = t \mathbf { i } + t ^ { 2 } \mathbf { j }$ . Find the normal component of the acceleration vector when t = 1.

A) $\frac { 1 } { \sqrt { 5 } }$

B) $\frac { 2 } { \sqrt { 5 } }$
C) $\frac { 3 } { \sqrt { 5 } }$
D) $\frac { 4 } { \sqrt { 5 } }$
E) $\sqrt { 5 }$
F) $\frac { 6 } { \sqrt { 5 } }$

G) $\frac { 7 } { \sqrt { 5 } }$

H) $\frac { 8 } { \sqrt { 5 } }$

Question 40

A cannon sits on top of a vertical tower 264 feet tall. It fires a cannonball at 80 ft/s. If the barrel of the cannon is elevated 30 degrees from the horizontal, find how far from the base of the tower the cannonball will land (assuming the ground around the tower is level).

Question 41

A helix has radius 5 and height 6, and makes 4 revolutions. Find parametric equations of this helix. What is the arc length of the helix?

Question 42

Find the unit normal vector N(t) to the curve r (t) = $\left\langle t , 2 t , t ^ { 2 } \right\rangle$ when t = 1.

A) $\left\langle \frac { 1 } { 3 } , \frac { 2 } { 3 } , \frac { 2 } { 3 } \right\rangle$

B) $( 1,0,0 )$
C) $( 0,1,0)$
D) $\left( 0 , \frac { 1 } { \sqrt { 2 } } , \frac { 1 } { \sqrt { 2 } } \right)$
E) $\left( - \frac { 2 } { 3 \sqrt { 5 } } , - \frac { 4 } { 3 \sqrt { 5 } } , \frac { 5 } { 3 \sqrt { 5 } } \right)$
F) $\left( \frac { 1 } { \sqrt { 2 } } , 0 , \frac { 1 } { \sqrt { 2 } } \right)$

G) $\left( \frac { 1 } { \sqrt { 2 } } , \frac { 1 } { \sqrt { 2 } } , 0 \right)$

H) $\left( \frac { 1 } { \sqrt { 2 } } , \frac { 1 } { \sqrt { 2 } } , \frac { 1 } { \sqrt { 2 } } \right)$

Question 43

Find the arc length of the curve given by

Question 44

Find the arc length of the curve given by

Question 45

Find the length of the curve $\mathbf { r } ( t ) = \left\langle \sin 2 t , \cos 2 t , 2 t ^ { \frac { 3 } { 2 } } \right\rangle 0 \leq t \leq 1$

A) $\frac { 2 } { 27 } ( 13 \sqrt { 13 } - 8 )$


B) $\frac { 13 } { 9 }$
C) $\frac { 13 \sqrt { 13 } - 6 } { 27 }$
D) $\frac { 16 } { 9 }$
E) $\frac { 10 } { 27 } ( \sqrt { 13 } - 2 )$
F) $\frac { 19 } { 9 }$

G) $\frac { 4 } { 9 } ( 7 \sqrt { 7 } - 2 )$

H) $\frac { 22 } { 9 }$

Question 46

Find the curvature $K$ of the curve $\mathbf { r } ( t ) = \langle \sin 2 t , 3 t , \cos 2 t \rangle \text { when } t = \frac { \pi } { 2 }$

A) $\frac { 3 } { 13 }$

B) $\frac { 4 } { 13 }$
C) $\frac { 6 } { 13 }$
D) $\frac { 8 } { 13 }$
E) $\frac { 1 } { 3 }$
F) $\frac { 4 } { 9 }$

G) $\frac { 2 } { 3 }$

H) $\frac { 8 } { 9 }$

Question 47

Find the length of the curve $\mathbf { r } ( t ) = \left\langle 2 t ^ { \frac { 3 } { 2 } } , 2 t + 1 , \sqrt { 5 } t \right\rangle 0 \leq t \leq 3$

A) $3 \sqrt { 21 }$
B)6
C)8
D)10
E)189
F)14
G)16
H)18

Question 48

Find the length of the circular helix described by

Question 49

Find the length of the curve $\mathbf { r } ( t ) = \langle 2 t , \sin t , \cos t \rangle , 0 \leq t \leq 2 \pi$

A) $2 \pi \sqrt { 2 }$

B) $\pi \sqrt { 10 }$
C) $2 \pi \sqrt { 3 }$
D) $\pi \sqrt { 14 }$
E) $4 \pi$
F)
$3 \pi \sqrt { 2 }$

G) $2 \pi \sqrt { 5 }$

H) $\pi \sqrt { 22 }$

Question 50

Find the unit normal vector N(t) to the curve r (t) = $\langle \sin t , t , \cos t \rangle$ when t = 0.

A) $( - 1,0,0 )$

B) $( 0,0 , - 1 )$
C) $( 1,0,0)$
D) $( 0,0,1 )$
E) $\left( 0 , \frac { 1 } { \sqrt { 2 } } , \frac { 1 } { \sqrt { 2 } } \right)$
F) $\left( \frac { 1 } { \sqrt { 2 } } , 0 , \frac { 1 } { \sqrt { 2 } } \right)$

G) $\left( \frac { 1 } { \sqrt { 2 } } , \frac { 1 } { \sqrt { 2 } } , 0 \right)$

H) $\left( \frac { 1 } { \sqrt { 3 } } , \frac { 1 } { \sqrt { 3 } } , \frac { 1 } { \sqrt { 3 } } \right)$

Question 51

Find the unit tangent vector T(t) to the curve r (t) = $\langle \sin t , t , \cos t \rangle$ when t = 0.

A) $( - 1,0,0 )$

B) $( 0,0 , - 1 )$
C) $( 1,0,0 )$
D) $( 0,0,1 )$
E) $\left( 0 , \frac { 1 } { \sqrt { 2 } } , \frac { 1 } { \sqrt { 2 } } \right)$
F) $\left( \frac { 1 } { \sqrt { 2 } } , 0 , \frac { 1 } { \sqrt { 2 } } \right)$

G) $\left( \frac { 1 } { \sqrt { 2 } } , \frac { 1 } { \sqrt { 2 } } , 0 \right)$

H) $\left( \frac { 1 } { \sqrt { 3 } } , \frac { 1 } { \sqrt { 3 } } , \frac { 1 } { \sqrt { 3 } } \right)$

Question 52

If , find the acceleration vector and the tangential component of the acceleration vector.

Question 53

Let . Show that the velocity vector is perpendicular to the acceleration vector.

Question 54

A particle is moving along the curve described by the parametric equations . Determine the velocity and acceleration vectors as well as the speed of the particle when t = 3.

Question 55

Let . Show that the acceleration vector is parallel to the normal vector N(t).

Question 56

Find the curvature $K$ of the curve $\mathbf { r } ( t ) = \left( t , t , 1 - t ^ { 2 } \right)$ at t = 0.

A)0
B) $\frac { 1 } { 8 }$
C) $\frac { 1 } { 4 }$
D) $\frac { 1 } { 2 }$
E)1
F)2
G)4
H)8

Question 57

Find the arc length of the curve given by

Question 58

A particle is traveling along a helix whose vector equation is given by , where . Find its maximum and minimum speeds.

Question 59

Find the curvature $K$ of the curve $y = 2 x ^ { 2 }$ at x = 0.

A)0
B) $\frac { 1 } { 8 }$
C) $\frac { 1 } { 4 }$
D) $\frac { 1 } { 2 }$
E)1
F)2
G)4
H)8

Question 60

Find the unit tangent vector T(t) to the curve r (t) = $\left( t ^ { 2 } - 1,3 t ^ { 2 } - t ^ { 4 } , \frac { 2 } { t } \right)$ when t = 1.

A) $\left( \frac { 1 } { \sqrt { 3 } } , \frac { 1 } { \sqrt { 3 } } , - \frac { 1 } { \sqrt { 3 } } \right)$

B) $\left( \frac { 1 } { \sqrt { 3 } } , - \frac { 1 } { \sqrt { 3 } } , - \frac { 1 } { \sqrt { 3 } } \right)$
C) $( 0,1,0)$
D) $( 0,0,1 )$
E) $\left( \frac { 1 } { \sqrt { 3 } } , \frac { 1 } { \sqrt { 3 } } , \frac { 1 } { \sqrt { 3 } } \right)$
F) $\left( \frac { 1 } { \sqrt { 2 } } , 0 , \frac { 1 } { \sqrt { 2 } } \right)$

G) $\left( \frac { 1 } { \sqrt { 2 } } , \frac { 1 } { \sqrt { 2 } } , 0 \right)$

H) $\left( \frac { 1 } { \sqrt { 2 } } , \frac { 1 } { \sqrt { 2 } } , \frac { 1 } { \sqrt { 2 } } \right)$

Question 61

Find the unit tangent and the unit normal to the graph of the vector function

Question 62

Find the tangent vector $\mathbf { r } ^ { \prime }$ (t) of the function r (t) = $\left( t , \sqrt { t } , \frac { 1 } { 2 \sqrt { t } } \right)$ when t = $\frac { 1 } { 4 }$ .

A) $( 1,2,4)$

B) $( 1,2 , - 2)$
C) $( 1,1,4 )$
D) $( 1,1 , - 2 )$
E) $( 1,2 , - 4 )$
F) $(1,2,2 )$

G) $( 1,1 , - 4 )$

H) $( 1,1,2 )$

Question 63

Find the equation of the osculating circle of the curve

Question 64

Find the curvature of the ellipse whose equation is given by

Question 65

Find the equation of the osculating circle of the ellipse whose equation is given by

Question 66

Find the equation of the osculating circle of the ellipse whose equation is given by

Question 67

Use the curvature formula to compute the curvature of a straight line y = mx + b.

Question 68

Find the curvature of the ellipse whose equation is given by

Question 69

Find the center of the osculating circle of the curve described by

Question 70

Consider . Determine graphically where the curvature is maximal and minimal.

Question 71

Find the derivative of the vector function r (t) = t i + sin t j when t = 0.

A)i
E)-i + j
B)j
F)i - j
C)-i
G)-i - j
D)-j
H)i + j

Question 72

Suppose C is the curve given by the vector function . Find the unit tangent vector, the unit normal vector, and the curvature of C at the point where t = 1.

Question 73

Find the center of the osculating circle of the parabola at the origin.

Question 74

Find the derivative of the vector function r (t) = $\left\langle t , 1 / t , e ^ { t } \right\rangle$ when t = 1.

A) $( 0,1,1 )$

B) $( 1,0,1 )$
C) $( 1,1 , e)$
D) $( 0,0 , e )$
E) $( - 1,1 , e )$
F) $( 1 , - 1 , e )$

G) $( 1,1 , - 1 )$

H) $( 1,1,1 )$

Question 75

Show that if and are parallel at some point on the curve described by , then the curvature at that point is 0. Give an example of a curve for which and are always parallel.

Question 76

Find the curvature of the curve

Question 77

Find the equation of the plane normal to

Question 78

At what point does the curve have minimum curvature? What is the minimum curvature?

Question 79

At what point does the curve have maximum curvature? What is the maximum curvature?

Question 80

Consider r (t), the vector function describing the curve shown below. Put the curvatures at A, B, and C in order from smallest to largest.