Question 1

Find a parametric representation for the surface consisting of that part of the cylinder $x ^ { 2 } + z ^ { 2 } = 1$ that lies between the planes $y = - 1$ and y = 3.

Accepted Answer

$x = \sin \theta$ , $y = y$ , $z = \cos \theta$ , $0 \leq \theta \leq 2 \pi$ , $- 1 \leq y \leq 3$

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.

Accepted Answer

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

Accepted Answer

The correct parametric representation for $z = x^2 + y^2$ is $x = r \cos \theta$, $y = r \sin \theta$, $z = r^2$, which corresponds to choice H. This is because $x^2 + y^2 = (r \cos \theta)^2 + (r \sin \theta)^2 = r^2(\cos^2 \theta + \sin^2 \theta) = r^2$, which matches the given surface equation.

Question 4

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

Accepted Answer

The answer of A picture of a circular cylinder with...

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

Accepted Answer

The given parametric equations describe a circular cylinder. The parameters $s$ and $t$ control the angle around the axis and the height along the axis, respectively, with a fixed radius of 3.

Question 6

Find a parametric representation for the surface consisting of that part of the hyperboloid $- x ^ { 2 } - y ^ { 2 } + z ^ { 2 } = 1 \text { that lies below the disk } \left\{ ( x , y ) \mid x ^ { 2 } + y ^ { 2 } \leq 4 \right\}$ .

Accepted Answer

The answer of Find a parametric representation for the surface...

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

Accepted Answer

The parametric equations $\mathbf { r } ( t ) = \{ 3 \cos t , 3 \sin t , 5 \}$ describe a circle of radius 3 in the xy-plane at a fixed z-coordinate of 5.

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

Accepted Answer

The given parametric equations describe a cone. The parameters $t$ and $s$ represent the radius and the angle in polar coordinates, respectively, with $t$ also being the height. This creates a shape where for every fixed value of $t$, a circle of radius $t$ is generated in the $xy$-plane, and as $t$ increases, these circles form a cone extending along the $z$-axis.

Question 9

Find a parametric representation for the surface consisting of that part of the hyperboloid $- x ^ { 2 } - y ^ { 2 } + z ^ { 2 } = 1$ that lies below the rectangle $[ - 1,1 ] \times [ - 3,3 ]$ .

Accepted Answer

The answer of Find a parametric representation for the surface...

Question 10

Identify the surface with the vector equation $\mathbf { r } ( u , v ) = \cos u \sin v \mathbf { i } + \sin u \sin v \mathbf { j } + \cos v \mathbf { k } , 0 \leq u \leq 2 \pi , 0 \leq v \leq \frac { \pi } { 2 }$ . (Hint: First consider $x ^ { 2 } + y ^ { 2 }$ .)

Accepted Answer

The answer of Identify the surface with the vector equation...

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

Accepted Answer

The parametric equations given describe a helix. The $x$ and $y$ components ($3\cos t$ and $3\sin t$, respectively) describe a circle of radius 3 in the $xy$-plane, while the $z$ component increases linearly with $t$, indicating a movement upwards or downwards along the $z$-axis as $t$ increases, thus forming a helix.

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

Accepted Answer

The given parametric equations represent a plane because the vector function $\mathbf{r}(t, s)$ linearly depends on the parameters $s$ and $t$, indicating a flat, two-dimensional surface extending infinitely in its dimensions.

Question 13

Find a parametric representation for the surface consisting of that part of the elliptic paraboloid $x + y ^ { 2 } + 2 z ^ { 2 } = 4$ that lies in front of the plane x = 0.

Accepted Answer

The answer of Find a parametric representation for the surface...

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

Accepted Answer

The given parametric equations represent a straight line because each component of the vector function $\mathbf{r}(t) = \langle 1 + t, 3t, 3 - 5t \rangle$ is a linear function of $t$, indicating a single direction of motion in space.

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) $( 1,0 , e )$

Accepted Answer

The velocity of the particle is given by the derivative of the position function $\mathbf{r}(t)$ with respect to time $t$, which is $\mathbf{v}(t) = \frac{d}{dt}\mathbf{r}(t) = \left\langle 2t, 2, e^t \right\rangle$. Substituting $t = 1$ gives $\mathbf{v}(1) = \left\langle 2(1), 2, e^1 \right\rangle = (2, 2, e)$.

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

Accepted Answer

The given parametric equations represent a sphere of radius 3. The parameters $s$ and $t$ can be seen as spherical coordinates, where $s$ represents the polar angle and $t$ the azimuthal angle. The equations describe all points in space at a fixed distance (3) from the origin, which is the definition of 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) $x = x , y = y , z = \sqrt { - x ^ { 2 } - 5 y ^ { 2 } - 1 }$

Accepted Answer

The correct parametric representation for the upper half of the ellipsoid $x^2 + 5y^2 + z^2 = 1$ involves solving for $z$ in terms of $x$ and $y$, which gives $z = \sqrt{1 - x^2 - 5y^2}$. This ensures $z$ is always positive (upper half) and satisfies the original equation.

Question 18

Identify the surface with the vector equation $\mathbf { r } ( u , v ) = ( 1 + 2 u + 3 v ) \mathbf { i } + ( 5 - u + 4 v ) \mathbf { j } + ( 3 + 5 u - 7 v ) \mathbf { k }$

Accepted Answer

The answer of 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

Accepted Answer

The answer of Identify the geometric object that is represented...

Question 20

Find a parametric representation for the surface consisting of that part of the plane z = x + 3 that lies inside the cylinder $x ^ { 2 } + y ^ { 2 } = 1$ .

Accepted Answer

The answer of Find a parametric representation for the surface...

Quiz 10: Vector Functions

Find a parametric representation for the surface consisting of that part of the cylinder x2+z2=1x ^ { 2 } + z ^ { 2 } = 1x2+z2=1 that lies between the planes y=−1y = - 1y=−1 and y = 3.

Find a parametric representation for the surface z=x2+y2z = x ^ { 2 } + y ^ { 2 }z=x2+y2

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

Identify the geometric object that is represented by parametric equations r(t,s) =⟨3cos⁡s,3sin⁡s,t⟩\mathbf { r } ( t , s ) = \langle 3 \cos s , 3 \sin s , t \rangler(t,s) =⟨3coss,3sins,t⟩ .

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

Identify the geometric object that is represented by parametric equations r(t,s) =⟨tcos⁡s,tsin⁡s,t⟩\mathbf { r } ( t , s ) = \langle t \cos s , t \sin s , t \rangler(t,s) =⟨tcoss,tsins,t⟩ .

Find a parametric representation for the surface consisting of that part of the hyperboloid −x2−y2+z2=1- x ^ { 2 } - y ^ { 2 } + z ^ { 2 } = 1−x2−y2+z2=1 that lies below the rectangle [−1,1]×[−3,3][ - 1,1 ] \times [ - 3,3 ][−1,1]×[−3,3] .

Identify the geometric object that is represented by parametric equations r(t) =⟨3cos⁡t,3sin⁡t,t⟩\mathbf { r } ( t ) = \langle 3 \cos t , 3 \sin t , t \rangler(t) =⟨3cost,3sint,t⟩ .

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

Find a parametric representation for the surface consisting of that part of the elliptic paraboloid x+y2+2z2=4x + y ^ { 2 } + 2 z ^ { 2 } = 4x+y2+2z2=4 that lies in front of the plane x = 0.

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

Let the position function of a particle be r(t) =⟨t2,2t,et⟩\mathbf { r } ( t ) = \left\langle t ^ { 2 } , 2 t , e ^ { t } \right\rangler(t) =⟨t2,2t,et⟩ . Find the velocity of the particle when t = 1.

Identify the geometric object that is represented by parametric equations r(t,s) =⟨3sin⁡scos⁡t,3sin⁡ssin⁡t,3cos⁡s⟩\mathbf { r } ( t , s ) = \langle 3 \sin s \cos t , 3 \sin s \sin t , 3 \cos s \rangler(t,s) =⟨3sinscost,3sinssint,3coss⟩ .

Find a parametric representation for the surface consisting of the upper half of the ellipsoid x2+5y2+z2=1x ^ { 2 } + 5 y ^ { 2 } + z ^ { 2 } = 1x2+5y2+z2=1 .

Identify the surface with the vector equation r(u,v)=(1+2u+3v)i+(5−u+4v)j+(3+5u−7v)k\mathbf { r } ( u , v ) = ( 1 + 2 u + 3 v ) \mathbf { i } + ( 5 - u + 4 v ) \mathbf { j } + ( 3 + 5 u - 7 v ) \mathbf { k }r(u,v)=(1+2u+3v)i+(5−u+4v)j+(3+5u−7v)k

Identify the geometric object that is represented by parametric equations r(t,s) =⟨scos⁡t,ssin⁡t,t⟩\mathbf { r } ( t , s ) = \langle s \cos t , s \sin t , t \rangler(t,s) =⟨scost,ssint,t⟩ .

Find a parametric representation for the surface consisting of that part of the plane z = x + 3 that lies inside the cylinder x2+y2=1x ^ { 2 } + y ^ { 2 } = 1x2+y2=1 .

Find a parametric representation for the surface consisting of that part of the cylinder $x ^ { 2 } + z ^ { 2 } = 1$ that lies between the planes $y = - 1$ and y = 3.

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

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

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

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

Find a parametric representation for the surface consisting of that part of the hyperboloid $- x ^ { 2 } - y ^ { 2 } + z ^ { 2 } = 1$ that lies below the rectangle $[ - 1,1 ] \times [ - 3,3 ]$ .

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

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

Find a parametric representation for the surface consisting of that part of the elliptic paraboloid $x + y ^ { 2 } + 2 z ^ { 2 } = 4$ that lies in front of the plane x = 0.

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

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.

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

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

Identify the surface with the vector equation $\mathbf { r } ( u , v ) = ( 1 + 2 u + 3 v ) \mathbf { i } + ( 5 - u + 4 v ) \mathbf { j } + ( 3 + 5 u - 7 v ) \mathbf { k }$

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

Find a parametric representation for the surface consisting of that part of the plane z = x + 3 that lies inside the cylinder $x ^ { 2 } + y ^ { 2 } = 1$ .