Deck 17: Parameterization and Vector Fields

Question 1

Find parametric equations for the line through the point (1, 5, 2)and parallel to the vector in which the particle is moving with speed 24 (the parameter t represents time).

Answer 1

x = 1 + 16 t, y = 5 + 16 t, and z = 2 - 8 t.

Question 2

Let f(x, y, z)= xy + 6yz + zx.Then f(2, 2, 3)= 46.
Give an equation to the tangent plane to xy + 6yz + zx = 46.

Answer 2

5x + 20y + 14z = 92

Question 3

Find a parametric equation for the line which passes through the point (5, 1, -1)and is parallel to the line .

Answer 3

Question 4

Which of the following equations give alternate parameterizations of the line L parameterized by $\vec { r } = ( 1 + 2 t ) \vec { i } + ( 2 + 2 t ) \vec { j } - ( 1 + 4 t ) \vec { k } ?$

A) $\vec { r } = - ( 1 + t ) \vec { i } - t \vec { j } + ( 3 + 2 t ) \vec { k }$
B) $\vec { r } = ( 3 - 2 t ) \vec { i } + ( 2 - 2 t ) \vec { j } + ( 3 - 4 t ) \vec { k }$
C) $\vec { r } = ( 2 + 3 t ) \vec { i } + ( 1 + 3 t ) \vec { j } + ( 1 - 6 t ) \vec { k }$

Question 5

Write down a parameterization of the line through the points (2, 2, 4)and (6, 4, 2).Select all that apply.

A) $\vec { r } = 2 \vec { i } + 2 \vec { j } + 4 \vec { k } + t ( 4 \vec { i } + 2 \vec { j } - 2 \vec { k } )$
B) $\vec { r } = 2 \vec { i } + 2 \vec { j } + 4 \vec { k } + t ( 4 \vec { i } - 2 \vec { j } - 2 \vec { k } )$
C) $\vec { r } = 2 \vec { i } + 2 \vec { j } + 4 \vec { k } + t ( 4 \vec { i } + 2 \vec { j } + 2 \vec { k } )$
D) $\vec { r } = 2 \vec { i } + 2 \vec { j } + 4 \vec { k } - t ( 4 \vec { i } + 2 \vec { j } - 2 \vec { k } )$
E) $\vec { r } = 2 \vec { i } + 2 \vec { j } + 4 \vec { k } + t ( 2 \vec { i } + \vec { j } - \vec { k } )$

Question 6

Consider the plane x - 4y + 4z = 5 and the line x = a + bt, y = 2 + 2t, z = 5 - t.
Find the values of a and b such that the line lies in the plane.

Question 7

Describe the similarities and differences between the following two curves. $\text { Curve 1: } \vec { r } ( t ) = ( 3 + 3 t ) \vec { i } + ( 1 - t ) \vec { j } + ( 3 + 4 t ) \vec { k } , - \infty \leq t \leq \infty \text {, }$ $\text { Curve 2: } \vec { r } ( t ) = \left( 3 + 3 t ^ { 2 } \right) \vec { i } + \left( 1 - t ^ { 2 } \right) \vec { j } + \left( 3 + 4 t ^ { 2 } \right) \vec { k } , - \infty \leq t \leq \infty \text {. }$

Question 8

Find a parameterization of a curve that looks like sin y = z when viewed from the x-axis, and looks like x = z² when viewed from the y-axis.See the shadows drawn on the planes in the following picture. What does the curve look like when viewed from the z-axis?

Question 9

Find a parameterization for the circle of radius 4 in the xz-plane, centered at the point (3, 0, -5).Select all that apply.

A) $x = 3 + 4 \cos t , y = 0 , z = 5 + 4 \sin t$
B) $\vec { r } = ( 3 \vec { i } - 5 \vec { k } ) + 4 ( \cos t \vec { i } + \sin t \vec { k } )$
C) $x = 3 - 4 \sin t , y = 0 , z = - 5 + 4 \cos t$
D) $\vec { r } = ( 5 \vec { i } - 3 \vec { k } ) + 4 ( \cos t \vec { i } + \sin t \vec { k } )$
E) $x = 3 + 4 \cos t , y = 0 , z = - 5 + 4 \sin t$

Question 10

What curve, C, is traced out by the parameterization $\vec { r } = 2 \vec { i } + ( \cos t ) \vec { j } + ( \sin t ) \vec { k }$ for 0 $\le$ t $\le$ 2 $\pi$ ?
Either give a very complete verbal description or sketch the curve (or both).

Question 11

The line through the points (2, 5, 25)and (12, 7, 23)can be parameterized by .
What value of t gives the point (42, 13, 17)?

Question 12

Give parameterizations for a circle of radius 2 in the plane, centered at origin, traversed anticlockwise.

A) $x = \cos t , y = \sin t , 0 \leq t \leq 2 \pi$
B) $x = 4 \cos t , y = 4 \sin t , 0 \leq t \leq 2 \pi$
C) $x = 2 \cos t , y = - 2 \sin t , \quad 0 \leq t \leq 2 \pi$
D) $x = 2 \cos t , y = 2 \sin t , 0 \leq t \leq 2 \pi$
E) $x = 2 \cos t , y = 2 \sin t , \quad 0 \leq t \leq \pi$

Question 13

Find a parameterization for the curve y⁶ = x⁷ in the xy-plane.Select all that apply.

A) $x = t ^ { 7 / 6 } , y = t$
B) $x = t ^ { 6 } , y = t ^ { 7 }$
C) $x = t ^ { 6 / 7 } , y = t$
D) $x = t ^ { 7 } , y = t ^ { 6 }$
E) $x=t, y=t^{6 / 7}$

Question 14

Suppose z = f(x, y), f(1, 3)= 5 and $\nabla f ( 1,3 ) = 4 \vec { i } + 5 \vec { j }$ the vector $- 4 \vec { i } + 5 \vec { j } + \vec { k }$ is perpendicular to the graph of f(x, y)at the point (1, 3).

Question 15

The equation parameterizes a line through the point (4, 3, 7).
What is the value of t at this point?

Question 16

Consider the plane x - 4y + -2z = 5 and the line x = a + bt, y = 2 + -2t, z = 2 - t.
Find the value of b such that the line is perpendicular to the plane.

Question 17

Find parametric equations for a line through the points, A = (-2, 5, 4)and B = (-2, 25, 9)so that the point A corresponds to t = 0 and the point B to t = 5.

Question 18

Consider the plane and the line with parametric equation Give a value of which makes the line parallel to the plane.(There are many possible answers.)

Question 19

Give parameterizations for a circle of radius 3 in 3-space perpendicular to the y-axis centered at (4, -2, 0).

A) $x = 4 + 3 \cos t , y = - 2 , z = 3 \sin t , \quad 0 \leq t \leq 2 \pi$
B) $x = 4 + 3 \cos t , y = - 2 , z = 3 \sin t , \quad 0 \leq t \leq \pi$
C) $x = 4 + 3 \cos t , y = 3 \sin t , z = - 2 , \quad 0 \leq t \leq 2 \pi$
D) $x = 4 + 3 \cos t , \quad z = 3 \sin t , \quad 0 \leq t \leq 2 \pi$
E) $x = 3 \cos t , y = - 2 , z = 3 \sin t , \quad 0 \leq t \leq 2 \pi$

Question 20

A child is sliding down a helical slide.Her position at time t after the start is given in feet by $\vec { r } = \cos t \vec { i } + \sin t \vec { j } + ( 12 - t ) \vec { k }$ .The ground is the xy-plane.
At time t = 2 $\pi$ , the child leaves the slide on the tangent to the slide at that point.What is the equation of the tangent line?

Question 21

Write a formula for a vector field whose vectors are parallel to the x-axis and point away from the y-axis, with magnitude inversely proportional to the cube of the distance from the x-axis.

Question 22

Find the coordinates of the point where the line tangent to the curve at the point (4, 16, 64)crosses the xy-plane.

Question 23

A particle moves at a constant speed along a line through P = (10,-20, 22)and Q = (22, -46, 46).Find a parametric equation for the line if the particle passes through P at time t = 3 and passes through Q at time t = 7.

Question 24

Calculate the length of the curve from x = -3 to x = 3.

Question 25

A particle moves with position vector $\vec { r } ( t ) = \ln t \vec { i } + t ^ { - 1 } \vec { j } + e ^ { - t } \vec { k }$ . Describe the movement of the particle as t $\rightarrow$ $\infty$ .

A)The particle will approach the positive x-axis asymptotically as t $\rightarrow$ $\infty$ .Also, since each component of $\vec { v } ( t )$ approaches 0 as t $\rightarrow$ $\infty$ , we expect the particle to approach the positive y-axis with slower and slower speed.
B)The particle will approach the positive y-axis asymptotically as t $\rightarrow$$\infty$ .Also, since each component of $\vec { v } ( t )$ approaches 0 as t $\rightarrow$ $\infty$ , we expect the particle to approach the positive x-axis with slower and slower speed.
C)The particle will approach the positive x-axis asymptotically as t $\rightarrow$ $\infty$ .Also, since each component of $\vec { v } ( t )$ approaches 0 as t $\rightarrow$ $\infty$ , we expect the particle to approach the positive x-axis with increasing speed.
D)The particle will approach the positive x-axis asymptotically as t $\rightarrow$$\infty$ .Also, since each component of $\vec { v } ( t )$ approaches 0 as t $\rightarrow$ $\infty$ , we expect the particle to approach the positive x-axis with slower and slower speed.
E)The particle will approach the positive x-axis asymptotically as t $\rightarrow$ $\infty$ .Also, since each component of $\vec { v } ( t )$ approaches 0 as t $\rightarrow$ $\infty$ , we expect the particle to approach the positive z-axis with slower and slower speed.

Question 26

The equation describes the motion of a particle moving on a circle.Assume x and y are in miles and t is in days.
What is the speed of the particle when it passes through the point (0, 2)?

Question 27

Let f(x, y)be a function that depends on only one of the variables, that is, of the form f(x, y)= g(x)or f(x, y)= g(y). Could the following picture be the gradient of f?

A)No
B)Yes
C)Not possible to say

Question 28

The parametric vector form of the position of a roller coaster is $\vec { r } ( t ) = 30 \sin ( t ) \vec { i } + 30 \cos ( t ) \vec { j } + 15 \cos ( t ) \vec { k }$ Answer the following questions about the ride.
(a)The scariest point of the ride is when it is traveling fastest.For which value of t > 0 does this occur first?
(b)Does the velocity vector of the roller coaster ever point directly downward?

Question 29

Match the vector field $\vec{F}(x, y)=x \vec{i}+y \vec{j}$ with the descriptions (a)-(d).

A)A swirling in a clockwise direction.
B)An attractive force field pointing toward the origin.
C)A repulsive force field pointing away from the origin.
D)A swirling in a counter-clockwise direction.

Question 30

A vector field is shown below. Find

Question 31

For the following vector field, identify which one of the following formulas could represent it. The scales in the x and y directions are the same.No reasons need be given.

A) $\vec { i } + x \vec { j }$
B) $x ^ { 2 } \vec { i } + x y \vec { j }$
C) $y \vec { i }$
D) $x \vec { i } + y \vec { j }$

Question 32

Answer the following as "true", "false" or "need more information".
If a particle moves with velocity , then the particle stops at the origin.

Question 33

The path of an object moving in xyz-space is given by .
The temperature at a point (x, y, z)in space is given by Calculate the directional derivative of f in the direction of at the point (12, 3, 8), where is the velocity vector of the object..

Question 34

Find a parameterization of the curve and use it to calculate the path length of this curve from (0, 0)to (1, 1).

Question 35

The path of an object moving in xyz-space is given by .
The temperature at a point (x, y, z)in space is given by Calculate .

Question 36

Let and let C be the helix parameterized by Find an expression for the outward pointing normal vector whose component is 0 at an arbitrary point ( sin(t), cos(t), t)of C.

Question 37

Sketch the vector fields $\vec { v } = x \vec { i }$

A) $\text { Neither is correct. }$
B)
C)

Question 38

An object moves with constant velocity in 3-space.It passes through (4, 0, 1)at time t = 1 and through (13, 6, -11)at time t = 4.Find its velocity vector.

Question 39

The figure below shows the contour map of a function z = f(x, y). Let $\vec{F}$ be the gradient vector field of f, i.e., $\vec{F}=\text { gradf }$ Which of the vector fields show $\vec { F } ?$

A)
B)
C)

Question 40

A particle moves at a constant speed along a line through P = (7,-14, 13)and Q = (19, -37, 37).Find a parametric equation for the line if:
The speed of the particle is 9 units per second and it is moving in the direction of .

Question 41

Let be a constant velocity field.
Find the flow line of that passes through the origin at time t = 2.

Question 42

Consider the plane $\vec { r } ( s , t ) = ( - 4 + s - 4 t ) \vec { i } + ( 5 - s + 4 t ) \vec { j } + ( 6 - 4 t - s ) \vec { k }$ Does it contain the point (-7, 8, -7)?

Question 43

Let S be a circular cylinder of radius 0.2, such that the center of one end is at the origin and the center of the other end is at the point (5, 0, 4).
Let P be the plane containing the base of the cylinder (i.e., the plane through the origin perpendicular to the axis of the cylinder).
In each case, give a parameterization and specify the range of values your parameters must take on.
(i)the circle in which the cylinder, S, cuts the plane, P.
(ii)the surface of the cylinder S.

Question 44

Let $\vec { v } _ { 1 } = 2 \vec { i } - 1 \vec { j } + \vec { k }$ and $\vec { v } _ { 2 } = 1 \vec { i } + \vec { j } + \vec { k }$ Find a parametric equation for the plane through the point (1, 2, -1)and containing the vectors $\vec { v } _ { 1 }$ and $\vec { v } _ { 2 }$ Select all that apply.

A) $x = 1 + 2 t + 1 s , y = 2 - 1 t + s , z = - 1 + t + s$
B) $x = 1 + 2 t - 1 s , y = 2 - 1 t - s , z = - 1 + t - s$
C) $x = - 1 + 2 t + 1 s , y = - 2 - 1 t + s , z = 1 + t + s$
D) $x = 1 - 2 t + 1 s , y = 2 + 1 t + s , z = - 1 - t + s$

Question 45

Suppose

Find a function f(x, y, z) of three variables with the property that the vectors in on a level surface of f (x, y, z) are perpendicular to the level surface of f(x, y, z) at each point.

Question 46

Find the parametric equation of the plane through the point (5, 2, 2)and parallel to the lines $\vec { r } ( t ) = ( 1 - 2 t ) \vec { i } + ( 5 + 2 t ) \vec { j } + ( 3 - 4 t ) \vec { k }$ and $\vec { s } ( t ) = ( 3 - 4 t ) \vec { i } + 4 t \vec { j } + ( 4 - 2 t ) \vec { k }$
Select all that apply.

A) $x = 5 - 2 u - 4 v , y = 2 + 2 u + 4 v , z = 2 - 4 u - 2 v$
B) $x = 5 - 2 u - 4 v , y = 2 + 2 u + 4 v , z = 2 + 4 u - 2 v$
C) $x = 5 - 2 u + 4 v , y = 2 + 2 u - 4 v , z = 2 - 4 u + 2 v$
D) $x = - 2 u - 4 v , y = 2 u + 4 v , z = - 4 u - 2 v$
E) $x = 5 - 2 u - 4 v , y = 2 + 2 u + 4 v , z = 2 - 4 u - 2 v$ .

Question 47

Let S be a circular cylinder of radius 0.2, such that the center of one end is at the origin and the center of the other end is at the point (5, 0, 4).
Find the xyz-equation of the plane, P, containing the base of the cylinder (i.e., the plane through the origin perpendicular to the axis of the cylinder).

Question 48

Match the surface with its parameterization below.

A) $\cos t \vec { i } + ( s + t ) \vec { j } + \sin t \vec { k }$
B) $s \vec { i } + ( s + t ) \vec { j } + t ^ { 2 } \vec { k }$
C) $s \vec { i } + s \cos t \vec { j } + s \sin t \vec { k }$
D) $s \vec { i } + \left( s + t ^ { 3 } \right) \vec { j } + ( s + t ) \vec { k }$

Question 49

Find parametric equations for the cylinder $y ^ { 2 } + z ^ { 2 } = 16$

A) $\vec { r } ( s , \theta ) = 4 \cos \overrightarrow { \theta i } + 4 \sin \theta \vec { j }$
B) $\vec { r } ( s , \theta ) = 4 \cos \theta \vec { i } + 4 \sin \theta \vec { j } + s \vec { k }$
C) $\vec { r } ( s , \theta ) = \vec { s } + 16 \cos \theta \vec { j } + 16 \sin \theta \vec { k }$
D) $\vec { r } ( s , \theta ) = \operatorname { si } + 4 \cos \theta \vec { j } + 4 \sin \theta \vec { k }$
E) $\vec { r } ( s , \theta ) = \operatorname { si } + 4 \cos \theta \vec { j } + 4 \cos \theta \vec { k }$

Question 50

Let and Find a vector which is perpendicular to and to find an equation of the plane through the point (1, 2, -1)and with normal vector perpendicular to both and .Express your answer in the form

Question 51

Find an equation for a vector field with all of the following properties.
• Defined at all points except (1, -5)
• All vectors have length 1.
• All vectors point away from the point (1, -5).

Question 52

Let S be a circular cylinder of radius 0.2, such that the center of one end is at the origin and the center of the other end is at the point (2, 0, 7).
Find two unit vectors and in the plane, P, containing the base of the cylinder (i.e., the plane through the origin perpendicular to the axis of the cylinder)which are perpendicular to each other.

Question 53

The following equations represent a curve or a surface.Select the best geometric description. (Note: $\rho$ , $\varphi$ , $\theta$ are spherical coordinates; r, $\theta$ , z are cylindrical coordinates.) $x = t , y = 2 t , z = 3 t , 0 \leq t \leq 10$

A)Part of a line through the origin
B)Part of a cylinder.
C)Part of a cone.
D)Disk

Question 54

Find parametric equations for the cylinder $49 x ^ { 2 } + 11 y ^ { 2 } = 539 , - 7 \leq z \leq 11$

A) $x = 11 \cos t , y = 7 \sin t , z = s , 0 \leq t \leq 2 \pi , - 7 \leq s \leq 11$
B) $$x = \sqrt { 11 } \cos t _ { 1 } y = 7 \sin t _ { 1 } \quad z = t _ { 1 } \quad 0 \leq t \leq 2 \pi _ { 1 } - 7 \leq s \leq 11$$
C) $$x = \sqrt { 7 } \cos t _ { 1 } y = 11 \sin t _ { 1 } \quad z = s , \quad 0 \leq t \leq 2 \pi , - 7 \leq s \leq 11$$
D) $x=\sqrt{11} \cos t, y=7 \sin t, \quad z=s, \quad 0 \leq t \leq \pi,-7 \leq s \leq 11$
E) $x=\sqrt{11} \cos t, y=7 \sin t, \quad z=s, \quad 0 \leq t \leq 2 \pi\,-7 \leq s \leq 11$

Question 55

Using cylindrical coordinates, find parametric equations for the cylinder $x ^ { 2 } + y ^ { 2 } = 16$ Select all that apply.

A) $x = 4 \cos \theta , y = 4 \sin \theta , z = z , 0 \leq \theta \leq 2 \pi , - \infty < z < \infty$
B) $x = 4 \cos \theta , y = 4 \sin \theta , 0 \leq \theta \leq 2 \pi , - \infty < z < \infty$
C) $x = 16 \cos \theta , y = 16 \sin \theta , z = z , \quad 0 \leq \theta \leq 2 \pi , - \infty < z < \infty$
D) $x = 4 \sin \theta , y = 4 \cos \theta , z = z , 0 \leq \theta \leq 2 \pi , - \infty < z < \infty$
E) $x = 16 \cos \theta , y = 16 \sin \theta , 0 \leq \theta \leq 2 \pi , - \infty < z < \infty$

Question 56

Consider the parametric surface $\vec { r } ( s , t ) = s \sin \left( \frac { \pi } { 2 } \right) \vec { i } + s \cos \left( \frac { \pi } { 2 } \right) \vec { j } + 4 t \vec { k }$ Does it contain the y-axis?

Question 57

The vector field represents an ocean current.An iceberg is at the point (1, 2)at t = 0.
Determine the position of the iceberg at time t = 2.

Question 58

Consider the parametric surface $\vec { r } ( s , t ) = s \sin \left( \frac { \pi } { 2 } \right) \vec { i } + s \cos \left( \frac { \pi } { 2 } \right) \vec { j } + 2 t \vec { k }$ Does it contain the point (0, -3, -4)?

Question 59

Consider the plane $\vec { r } ( s , t ) = ( - 4 + s - 5 t ) \vec { i } + ( 5 - s + 5 t ) \vec { j } + ( 2 - 10 t + s ) \vec { k }$ Find a normal vector to the plane.

A) $\frac { 1 } { \sqrt { 2 } } ( \vec { i } + \vec { k } )$
B) $\frac { 1 } { \sqrt { 3 } } ( \vec { i } + \vec { j } - \vec { k } )$
C) $\frac { 1 } { \sqrt { 2 } } ( \vec { i } + \vec { j } )$
D) $\frac { 1 } { \sqrt { 3 } } ( \vec { i } + \vec { j } + \vec { k } )$
E) $\frac { 1 } { \sqrt { 2 } } ( \vec { i } - \vec { j } )$

Question 60

Find parametric equations for the sphere

Question 61

Consider the curve $\vec { r } ( t ) = \left( 2 t ^ { 2 } + 1 \right) \vec { i } + \left( t ^ { 2 } - 2 \right) \vec { j } + t \vec { k }$ Does it pass through the point (1, -2, 0)?

Question 62

Parameterize the circle $x ^ { 2 } + y ^ { 2 } = 25 , z = 15$ Select all that apply.

A) $x = 5 \cos t , y = 5 \sin t , z = 15$
B) $x = 5 \sin t , y = 5 \cos t , z = 15$
C) $x = - 5 \cos t , y = - 5 \sin t , z = 15$
D) $x = 5 \cos 2 t , y = 5 \sin 2 t , z = 15$
E) $x = 25 \cos t , y = 25 \sin t , z = 15$

Question 63

Find the parametric equations for the line of the intersection of the planes and .

Question 64

Use cylindrical coordinates to parameterize the part of the plane x + y - z = 10 inside the cylinder $x ^ { 2 } + y ^ { 2 } = 4$ .

Question 65

If a particle is moving along a parameterized curve $\vec { r } ( t )$ , then the acceleration vector at any point cannot be parallel to the velocity vector at that point.

Question 66

Are the lines parallel? $l _ { 1 } : x = 2 t + 5 , y = 3 t + 3 , z = - 4 t - 2$ $l _ { 2 } : x = 5 t + 1 , y = 2 t - 4 , z = 11 t + 7$

Question 67

The lines and are perpendicular.Find a.

Question 68

Consider the curve $\vec { r } ( t ) = \left( 2 t ^ { 2 } + 1 \right) \vec { i } + \left( t ^ { 2 } + 1 \right) \vec { j } + t \vec { k }$ Does the curve lie on the parametric surface $x = s ^ { 2 } + t ^ { 2 } , y = s ^ { 2 } , z = t ?$

Question 69

Let S be the parametric surface for , .
(a)What does the projection of S onto the xy-plane look like?
(b)Show that S is part of the surface .
(c)Find a unit vector that is normal to the surface at the point (1,2,3).

Question 70

The curve passes through the point (-12, 1, 51).Find a and b.

Question 71

Consider the curve .
(a)Find a unit vector tangent to the curve at the point (1,2,3).
(b)Show that the curve lies on the surface .

Question 72

Do the lines intersect? $l _ { 1 } : x = - 5 t + 1 , y = - 7 t + 1 , z = 3 t$ $l _ { 2 } : x = 12 t + 3 , y = 7 , z = - 11 t - 6$

Question 73

A surveyor wants to measure the height of a building.At point A = ( 733, -369, 0)on the ground, she observes that the vector is parallel to , where C is the highest point of the building.At point B = (418, 471, 0)she observes that the vector is parallel to .Given that ground level is the plane z = 0 and the units are feet, find the height of the building.

Question 74

If a particle moves with constant speed, the path of the particle must be a line.

Question 75

Parameterize the curve which lies on the plane 5x - 10y + z = 6 above the circle $x ^ { 2 } + y ^ { 2 } = 25$

A) $\vec { r } ( t ) = 5 \sin t \vec { i } + 5 \cos t \vec { j } + ( 6 - 5 \sin t + 10 \cos t ) \vec { k }$
B) $\vec { r } ( t ) = 5 \sin t \vec { i } + 5 \cos t \vec { j } + ( 6 - 25 \sin t - 50 \cos t ) \vec { k }$
C) $\vec { r } ( t ) = 5 \sin t \vec { i } + 5 \cos t \vec { j } + ( 6 + 25 \sin t + 25 \cos t ) \vec { k }$
D) $\vec { r } ( t ) = 5 \sin t \vec { i } + 5 \cos t \vec { j } + ( 6 - 25 \sin t + 50 \cos t ) \vec { k }$
E) $\vec { r } ( t ) = 5 \sin t \vec { i } + 5 \cos t \vec { j } + ( 6 - 5 \sin t + 25 \cos t ) \vec { k }$

Question 76

Consider the curve Find the equation of the tangent line at the point where t = 2.

Question 77

Use spherical coordinates to parameterize the part of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ above the plane z = 1.

Question 78

The two planes are parallel.
Plane 1: $x = 2 + s + t , y = 4 + s - t , z = 1 + 2 s$
Plane 2: $x = 2 + s + 2 t , y = t , z = s - t$

Question 79

Suppose the vector field $\vec { F } ( x , y )$ represents an ocean current and that $\vec { r } ( t ) = 8 \cos t \vec { i } + 9 t \sin t \vec { j }$ is a flow line of $\vec{F}$ Find the acceleration vector of the flow line at (0, 9 $\pi$ /2).

Question 80

How many parameters are needed to parameterize a surface in 3-space?