Question 1

Describe the surface whose equation in cylindrical coordinates is $ \theta $ = 3.

Accepted Answer

A) Cylinder with vertical axis 
B) Cylinder with horizontal axis 
C) Sphere 
D) Vertical plane or half-plane 
E) Horizontal plane or half-plane
F)Paraboloid
G)Cone or half-cone with vertical axis
H)Cone or half-cone with horizontal axis 
A) Cylinder with vertical axis 
B) Cylinder with horizontal axis 
C) Sphere 
D) Vertical plane or half-plane 
E) Horizontal plane or half-plane
F)Paraboloid
G)Cone or half-cone with vertical axis
H)Cone or half-cone with horizontal axis

Question 2

Find an equation of a plane containing the point P(2, -1, 1) and the z-axis.

Accepted Answer

A)  $y + z = 0$ 
B)  $x - y = 3$ 
C)  $x = 2$ 
D)  $x - 2 z = 0$ 
E)  $y - z = 0$ 
F) $x + 2 y = 0$ 
G) $z = 1$ 
H) $z = - 1$ 
A)  $y + z = 0$ 
B)  $x - y = 3$ 
C)  $x = 2$ 
D)  $x - 2 z = 0$ 
E)  $y - z = 0$ 
F) $x + 2 y = 0$ 
G) $z = 1$ 
H) $z = - 1$

Question 3

What does the fact that $\mathbf { a } \cdot \mathbf { a }$ = 0 imply about the vector a?

Accepted Answer

The answer of What does the fact that \(\mathbf {...

Question 4

Let f(x, y) = $4 - y ^ { 2 }$ (a) Evaluate $f ( - 1 , - 1 )$ .(b) Find the domain of f.(c) Find the range of f.

Accepted Answer

(a) 3
(b)

Question 5

Explain why there is no vector a such that $\langle 1,0,4 \rangle \times \mathbf { a } = \langle 0,1,1 \rangle$

Accepted Answer

@#IMG-DLM& which imp

Question 6

Find the length of the vector $2 a + 3 b$ , where $\mathbf { a } = \langle 1,1 \rangle$ and $\mathbf { b } = \langle - 1,2 \rangle$ .

Accepted Answer

A)  $\sqrt { 51 }$ 
B)  $\sqrt { 53 }$ 
C)  $\sqrt { 57 }$ 
D)  $\sqrt { 65 }$ 
E)  $\sqrt { 71 }$ 
F) $\sqrt { 73 }$ 
G) $\sqrt { 82 }$ 
H) $\sqrt { 87 }$ 
A)  $\sqrt { 51 }$ 
B)  $\sqrt { 53 }$ 
C)  $\sqrt { 57 }$ 
D)  $\sqrt { 65 }$ 
E)  $\sqrt { 71 }$ 
F) $\sqrt { 73 }$ 
G) $\sqrt { 82 }$ 
H) $\sqrt { 87 }$

Question 7

Let a = $\langle 0,1,2 \rangle$ , b = $\langle- 1 , - 1,3 \rangle$ , c = $\langle 2,4 , - 2 \rangle$ , and d = $\langle 1,3 , - 3 \rangle$ . Find $(\mathbf{a} \cdot \mathbf{d}) \mathbf{b}-(\mathbf{b} \cdot \mathbf{c}) \mathbf{d}$ .

Accepted Answer

The answer of Let a = $\langle 0,1,2 \rangle$ ,...

Question 8

For which values of t are a = $\langle t + 2 , t , t\rangle$ and b = $\langle t - 2 , t + 1 , t \rangle$ orthogonal?

Accepted Answer

The answer of For which values of t are a...

Question 9

Identify the trace of the surface $x = y ^ { 2 } + z ^ { 2 }$ in the plane y = 0.

Accepted Answer

A) ellipse but not a circle 
B) parabola 
C) hyperbola 
D) circle 
E) two parallel straight lines
F)two intersecting straight lines
G)point
H)straight line 
A) ellipse but not a circle 
B) parabola 
C) hyperbola 
D) circle 
E) two parallel straight lines
F)two intersecting straight lines
G)point
H)straight line

Question 10

Let f(x, y) = $\sqrt { 4 - x ^ { 2 } }$ (a) Evaluate $f ( - 1 , - 1 )$ .(b) Find the domain of f.(c) Find the range of f.

Accepted Answer

(a) @#IMG-DLM& (b) @#IMG-DLM&

Question 11

Do the two lines $\mathrm { x } _ { 1 } ( t ) = \langle 1,1,3 \} + t ( - 1,0,2 \}$ and $\mathrm { x } _ { 2 } = \langle - 1,1,4 \rangle + s \langle 2,0,1 \}$ intersect? If so, find the point of intersection.

Accepted Answer

The answer of Do the two lines \(\mathrm { x...

Question 12

Find the domain of the function f(x, y) = $\sqrt { x - y ^ { 2 } }$ .

Accepted Answer

A) All points on or to the left of $x = y ^ { 2 }$ e.All points on or to the left of x = 0 
B) All points on or to the right of $x = y ^ { 2 }$ f.All points on or to the right of x = 0 
C) All points to the left of $x = y ^ { 2 }$ g.All points to the left of x = 0 
D) All points to the right of $x = y ^ { 2 }$ h.All points in the xy-plane 
A) All points on or to the left of $x = y ^ { 2 }$ e.All points on or to the left of x = 0 
B) All points on or to the right of $x = y ^ { 2 }$ f.All points on or to the right of x = 0 
C) All points to the left of $x = y ^ { 2 }$ g.All points to the left of x = 0 
D) All points to the right of $x = y ^ { 2 }$ h.All points in the xy-plane

Question 13

Let f(x, y) = $x ^ { 2 } + 2 x y + y ^ { 2 }$ . If x = 2, find f(x, 2x).

Accepted Answer

A) 12 
B) 16 
C) 24 
D) 28 
E) 32
F)36
G)42
H)48 
A) 12 
B) 16 
C) 24 
D) 28 
E) 32
F)36
G)42
H)48

Question 14

Let a be the vector shown below. Find the horizontal component b and the vertical component c of a if a = 3.

Accepted Answer

The answer of Let a be the vector shown below....

Question 15

Find the volume of the parallelepiped spanned by the vectors 2i + 3j + 5k, 3i - j + 4k, and -i - 2j + 3k.

Accepted Answer

The answer of Find the volume of the parallelepiped spanned...

Question 16

Identify the graph of the function f(x, y) = $3 - x ^ { 2 } - y ^ { 2 }$ .

Accepted Answer

A) Cone 
B) Paraboloid 
C) Ellipsoid 
D) Hyperboloid of one sheet 
E) Hyperboloid of two sheets
F)Hyperbolic cylinder
G)Elliptic cylinder
H)Parabolic cylinder 
A) Cone 
B) Paraboloid 
C) Ellipsoid 
D) Hyperboloid of one sheet 
E) Hyperboloid of two sheets
F)Hyperbolic cylinder
G)Elliptic cylinder
H)Parabolic cylinder

Question 17

Find the coordinate of the terminal point of a vector $\mathbf { a } = \langle 3 , - 2,2 \}$ whose initial point is $( 1 , - 1,3 )$ .

Accepted Answer

A)  $( - 1,1 , - 3 )$ 
B)  $( - 4,3 , - 5 )$ 
C)  $( - 3,2 , - 2 )$ 
D)  $( - 5,4 , - 8 )$ 
E)  $( 1 , - 1,3 )$ 
F) $( 4 , - 3,5 )$ 
G) $( 3 , - 2,2 )$ 
H) $( 5 , - 4,8 )$ 
A)  $( - 1,1 , - 3 )$ 
B)  $( - 4,3 , - 5 )$ 
C)  $( - 3,2 , - 2 )$ 
D)  $( - 5,4 , - 8 )$ 
E)  $( 1 , - 1,3 )$ 
F) $( 4 , - 3,5 )$ 
G) $( 3 , - 2,2 )$ 
H) $( 5 , - 4,8 )$

Question 18

Let a and b be vectors such that $| \mathbf { a } |$ = 2 and $| \mathbf { b } |$ = 3. Assume that the angle between a and b is $\frac { \pi } { 3 }$ . Find $| ( 2 \mathbf { a } + 3 \mathbf { b } ) \times ( \mathbf { a } + 4 \mathbf { b } ) |$ .

Accepted Answer

The answer of Let a and b be vectors such...

Question 19

Find the area of quadrilateral ABCD. Note that ABCD is not a parallelogram.

Accepted Answer

The answer of Find the area of quadrilateral ABCD. Note...

Question 20

Find the coordinate of the terminal point of a vector $\mathbf { a } = \langle 4 , - 3 \rangle$ whose initial point is $( 1,1 )$ .

Accepted Answer

A)  $( 1,1 )$ 
B)  $( 3 , - 4 )$ 
C)  $( 4 , - 3 )$ 
D)  $( 5 , - 2 )$ 
E)  $( - 1 , - 1 )$ 
F) $( - 3,4 )$ 
G) $( - 4,3 )$ 
H) $( - 5,2 )$ 
A)  $( 1,1 )$ 
B)  $( 3 , - 4 )$ 
C)  $( 4 , - 3 )$ 
D)  $( 5 , - 2 )$ 
E)  $( - 1 , - 1 )$ 
F) $( - 3,4 )$ 
G) $( - 4,3 )$ 
H) $( - 5,2 )$

Describe the surface whose equation in cylindrical coordinates is $\theta$ = 3.

Find an equation of a plane containing the point P(2, -1, 1) and the z-axis.

What does the fact that $\mathbf { a } \cdot \mathbf { a }$ = 0 imply about the vector a?

Let f(x, y) = $4 - y ^ { 2 }$ (a) Evaluate $f ( - 1 , - 1 )$ .(b) Find the domain of f.(c) Find the range of f.

Explain why there is no vector a such that $\langle 1,0,4 \rangle \times \mathbf { a } = \langle 0,1,1 \rangle$

Find the length of the vector $2 a + 3 b$ , where $\mathbf { a } = \langle 1,1 \rangle$ and $\mathbf { b } = \langle - 1,2 \rangle$ .

Let a = $\langle 0,1,2 \rangle$ , b = $\langle- 1 , - 1,3 \rangle$ , c = $\langle 2,4 , - 2 \rangle$ , and d = $\langle 1,3 , - 3 \rangle$ . Find $(\mathbf{a} \cdot \mathbf{d}) \mathbf{b}-(\mathbf{b} \cdot \mathbf{c}) \mathbf{d}$ .

For which values of t are a = $\langle t + 2 , t , t\rangle$ and b = $\langle t - 2 , t + 1 , t \rangle$ orthogonal?

Identify the trace of the surface $x = y ^ { 2 } + z ^ { 2 }$ in the plane y = 0.

Let f(x, y) = $\sqrt { 4 - x ^ { 2 } }$ (a) Evaluate $f ( - 1 , - 1 )$ .(b) Find the domain of f.(c) Find the range of f.

Do the two lines $\mathrm { x } _ { 1 } ( t ) = \langle 1,1,3 \} + t ( - 1,0,2 \}$ and $\mathrm { x } _ { 2 } = \langle - 1,1,4 \rangle + s \langle 2,0,1 \}$ intersect? If so, find the point of intersection.

Find the domain of the function f(x, y) = $\sqrt { x - y ^ { 2 } }$ .

Let f(x, y) = $x ^ { 2 } + 2 x y + y ^ { 2 }$ . If x = 2, find f(x, 2x).

Let a be the vector shown below. Find the horizontal component b and the vertical component c of a if a = 3.

Find the volume of the parallelepiped spanned by the vectors 2i + 3j + 5k, 3i - j + 4k, and -i - 2j + 3k.

Identify the graph of the function f(x, y) = $3 - x ^ { 2 } - y ^ { 2 }$ .

Find the coordinate of the terminal point of a vector $\mathbf { a } = \langle 3 , - 2,2 \}$ whose initial point is $( 1 , - 1,3 )$ .

Let a and b be vectors such that $| \mathbf { a } |$ = 2 and $| \mathbf { b } |$ = 3. Assume that the angle between a and b is $\frac { \pi } { 3 }$ . Find $| ( 2 \mathbf { a } + 3 \mathbf { b } ) \times ( \mathbf { a } + 4 \mathbf { b } ) |$ .

Find the area of quadrilateral ABCD. Note that ABCD is not a parallelogram.

Find the coordinate of the terminal point of a vector $\mathbf { a } = \langle 4 , - 3 \rangle$ whose initial point is $( 1,1 )$ .

Functions and Models

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Integrals

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Infinite Sequences and Series

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Multiple Integrals

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Exam 9: Vectors and the Geometry of Space

Describe the surface whose equation in cylindrical coordinates is θ \theta θ = 3.

Find an equation of a plane containing the point P(2, -1, 1) and the z-axis.

What does the fact that a⋅a\mathbf { a } \cdot \mathbf { a }a⋅a = 0 imply about the vector a?

Let f(x, y) = 4−y24 - y ^ { 2 }4−y2 (a) Evaluate f(−1,−1)f ( - 1 , - 1 )f(−1,−1) .(b) Find the domain of f.(c) Find the range of f.

Explain why there is no vector a such that ⟨1,0,4⟩×a=⟨0,1,1⟩\langle 1,0,4 \rangle \times \mathbf { a } = \langle 0,1,1 \rangle⟨1,0,4⟩×a=⟨0,1,1⟩

Find the length of the vector 2a+3b2 a + 3 b2a+3b , where a=⟨1,1⟩\mathbf { a } = \langle 1,1 \ranglea=⟨1,1⟩ and b=⟨−1,2⟩\mathbf { b } = \langle - 1,2 \rangleb=⟨−1,2⟩ .

For which values of t are a = ⟨t+2,t,t⟩\langle t + 2 , t , t\rangle⟨t+2,t,t⟩ and b = ⟨t−2,t+1,t⟩\langle t - 2 , t + 1 , t \rangle⟨t−2,t+1,t⟩ orthogonal?

Identify the trace of the surface x=y2+z2x = y ^ { 2 } + z ^ { 2 }x=y2+z2 in the plane y = 0.

Let f(x, y) = 4−x2\sqrt { 4 - x ^ { 2 } }4−x2​ (a) Evaluate f(−1,−1)f ( - 1 , - 1 )f(−1,−1) .(b) Find the domain of f.(c) Find the range of f.

Find the domain of the function f(x, y) = x−y2\sqrt { x - y ^ { 2 } }x−y2​ .

Let f(x, y) = x2+2xy+y2x ^ { 2 } + 2 x y + y ^ { 2 }x2+2xy+y2 . If x = 2, find f(x, 2x).

Let a be the vector shown below. Find the horizontal component b and the vertical component c of a if a = 3.

Find the volume of the parallelepiped spanned by the vectors 2i + 3j + 5k, 3i - j + 4k, and -i - 2j + 3k.

Identify the graph of the function f(x, y) = 3−x2−y23 - x ^ { 2 } - y ^ { 2 }3−x2−y2 .

Find the coordinate of the terminal point of a vector a=⟨3,−2,2}\mathbf { a } = \langle 3 , - 2,2 \}a=⟨3,−2,2} whose initial point is (1,−1,3)( 1 , - 1,3 )(1,−1,3) .

Find the area of quadrilateral ABCD. Note that ABCD is not a parallelogram.

Find the coordinate of the terminal point of a vector a=⟨4,−3⟩\mathbf { a } = \langle 4 , - 3 \ranglea=⟨4,−3⟩ whose initial point is (1,1)( 1,1 )(1,1) .

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Differential Equations

Infinite Sequences and Series

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Filters

Describe the surface whose equation in cylindrical coordinates is $\theta$ = 3.

What does the fact that $\mathbf { a } \cdot \mathbf { a }$ = 0 imply about the vector a?

Let f(x, y) = $4 - y ^ { 2 }$ (a) Evaluate $f ( - 1 , - 1 )$ .(b) Find the domain of f.(c) Find the range of f.

Explain why there is no vector a such that $\langle 1,0,4 \rangle \times \mathbf { a } = \langle 0,1,1 \rangle$

Find the length of the vector $2 a + 3 b$ , where $\mathbf { a } = \langle 1,1 \rangle$ and $\mathbf { b } = \langle - 1,2 \rangle$ .

For which values of t are a = $\langle t + 2 , t , t\rangle$ and b = $\langle t - 2 , t + 1 , t \rangle$ orthogonal?

Identify the trace of the surface $x = y ^ { 2 } + z ^ { 2 }$ in the plane y = 0.

Let f(x, y) = $\sqrt { 4 - x ^ { 2 } }$ (a) Evaluate $f ( - 1 , - 1 )$ .(b) Find the domain of f.(c) Find the range of f.

Find the domain of the function f(x, y) = $\sqrt { x - y ^ { 2 } }$ .

Let f(x, y) = $x ^ { 2 } + 2 x y + y ^ { 2 }$ . If x = 2, find f(x, 2x).

Identify the graph of the function f(x, y) = $3 - x ^ { 2 } - y ^ { 2 }$ .

Find the coordinate of the terminal point of a vector $\mathbf { a } = \langle 3 , - 2,2 \}$ whose initial point is $( 1 , - 1,3 )$ .

Find the coordinate of the terminal point of a vector $\mathbf { a } = \langle 4 , - 3 \rangle$ whose initial point is $( 1,1 )$ .