Question 1

If $ p=(1,1, \sqrt{2}) $ in rectangular coordinates, find the spherical coordinates of P.

Accepted Answer

The answer of If $ p=(1,1, \sqrt{2}) $ in rectangular...

Question 2

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

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 3

A person pushing a lawnmower exerts a force of 30 pounds in the direction of the handle, which makes an angle of 30° with the ground. How much work is done in moving the lawnmower 20 feet?

Accepted Answer

A) 300 $\sqrt { 3 }$ e.100 
B)  $\frac { 300 } { \sqrt { 3 } }$ f.200 
C) 200 $\sqrt { 3 }$ g.300 
D)  $\frac { 200 } { \sqrt { 3 } }$ h.400 
A) 300 $\sqrt { 3 }$ e.100 
B)  $\frac { 300 } { \sqrt { 3 } }$ f.200 
C) 200 $\sqrt { 3 }$ g.300 
D)  $\frac { 200 } { \sqrt { 3 } }$ h.400

Question 4

Find an equation of the plane containing P(-1, 2, 1) and the line $\frac { x + 1 } { 2 } = \frac { y } { 5 } = \frac { z - 3 } { 1 }$ .

Accepted Answer

The answer of Find an equation of the plane containing...

Question 5

Let $\mathbf { a } = \langle 5 , - 1 , - 2 \}$ and $\mathbf { b } = \{ 1 , - 4,3 \}$ .(a) Compute $a + 2 b$ .(b) Find $| \mathbf { a } |$ .(c) Find the unit vector in the direction of b.

Accepted Answer

The answer of Let \(\mathbf { a } = \langle...

Question 6

Given a = $\langle 7,4 \rangle$ , b = $\langle 1,2 \rangle$ , find the vector projection of b onto a.

Accepted Answer

A)  $\langle 1,2 \rangle$ 
B)  $\langle 3,6 \rangle$ 
C)  $\langle - 1 , - 2 \rangle$ 
D)  $\langle - 3 , - 6 \rangle$ 
E)  $\langle 7,4\rangle$ 
F) $\langle \frac { 21 } { 13 } , \frac { 12 } { 13 } \rangle$ 
G) $\langle - 7 , - 4 \rangle$ 
H) $\langle - \frac { 21 } { 13 } , - \frac { 12 } { 13 } \rangle$ 
A)  $\langle 1,2 \rangle$ 
B)  $\langle 3,6 \rangle$ 
C)  $\langle - 1 , - 2 \rangle$ 
D)  $\langle - 3 , - 6 \rangle$ 
E)  $\langle 7,4\rangle$ 
F) $\langle \frac { 21 } { 13 } , \frac { 12 } { 13 } \rangle$ 
G) $\langle - 7 , - 4 \rangle$ 
H) $\langle - \frac { 21 } { 13 } , - \frac { 12 } { 13 } \rangle$

Question 7

Determine all values for a such that the vectors $\mathbf { x } = 2 \mathbf { i } + \mathbf { j } + 2 \mathbf { k }$ and $\mathbf { y } = \mathbf { i } + 2 \mathbf { j } + a \mathbf { k }$ will form a 60° angle.

Accepted Answer

The answer of Determine all values for a such that...

Question 8

If vector a = 3i + 4j - 2k; and vector b = 2i - j + 5k, find a vector perpendicular to both vectors.

Accepted Answer

Any scalar multiple

Question 9

Find a vector of length 6 with the same direction as $\mathbf { a } = \langle 2,2 , - 1 \rangle$

Accepted Answer

The answer of Find a vector of length 6 with...

Question 10

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

Accepted Answer

A) All points on or to the left of $x = y ^ { 2 }$ 
B) All points on or to the right of $x = y ^ { 2 }$ 
C) All points to the left of $x = y ^ { 2 }$ 
D) All points to the right of $x = y ^ { 2 }$ 
E) All points on or to the left of x = 0
F)All points on or to the right of x = 0
G)All points to the left of x = 0
H)All points in the xy-plane 
A) All points on or to the left of $x = y ^ { 2 }$ 
B) All points on or to the right of $x = y ^ { 2 }$ 
C) All points to the left of $x = y ^ { 2 }$ 
D) All points to the right of $x = y ^ { 2 }$ 
E) All points on or to the left of x = 0
F)All points on or to the right of x = 0
G)All points to the left of x = 0
H)All points in the xy-plane

Question 11

Identify the trace of the surface $x ^ { 2 } = y ^ { 2 } + z ^ { 2 }$ in the plane z = 1.

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 12

For vectors a and b, given that $| \mathbf { a } \times \mathbf { b } | = | \mathbf { a } | | \mathbf { b } |$ , which of the following statements is true?

Accepted Answer

A) a and b are parallel 
B) a and b are not parallel 
C) a = b 
D) a and b are perpendicular 
E) a and b are not perpendicular 
A) a and b are parallel 
B) a and b are not parallel 
C) a = b 
D) a and b are perpendicular 
E) a and b are not perpendicular

Question 13

Sketch the graph of $z = x ^ { 2 } + y ^ { 2 }$ in $\mathbb { R } ^ { 3 }$ , and name the surface.

Accepted Answer

The answer of Sketch the graph of \(z = x...

Question 14

Describe the vertical traces x = 0 and the horizontal traces z = -1 (if any) for the surfaces $z = x ^ { 2 } + y ^ { 2 }$ and $z ^ { 2 } = x ^ { 2 } + y ^ { 2 }$ .

Accepted Answer

@#IMG-DLM& : x = 0 @#IMG-DLM& @#IMG-DLM& , a parab

Question 15

Find a unit vector that has the same direction as the vector $\langle6 , - 8 \rangle$ .

Accepted Answer

A)  $\langle \frac { 4 } { 5 } , \frac { 3 } { 5 } \rangle$ 
B)  $\left( \frac { 3 } { 5 } , \frac { 4 } { 5 } \right)$ 
C)  $\left( \frac { 1 } { 3 } , \frac { 1 } { 4 } \right)$ 
D)  $( 4 , - 3 )$ 
E)  $\left( - \frac { 3 } { 5 } , - \frac { 4 } { 5 } \right)$ 
F) $\left( - \frac { 3 } { 5 } , \frac { 4 } { 5 } \right)$ 
G) $\left( \frac { 3 } { 5 } , - \frac { 4 } { 5 } \right)$ 
H) $\langle 6 , - 8 \rangle$ 
A)  $\langle \frac { 4 } { 5 } , \frac { 3 } { 5 } \rangle$ 
B)  $\left( \frac { 3 } { 5 } , \frac { 4 } { 5 } \right)$ 
C)  $\left( \frac { 1 } { 3 } , \frac { 1 } { 4 } \right)$ 
D)  $( 4 , - 3 )$ 
E)  $\left( - \frac { 3 } { 5 } , - \frac { 4 } { 5 } \right)$ 
F) $\left( - \frac { 3 } { 5 } , \frac { 4 } { 5 } \right)$ 
G) $\left( \frac { 3 } { 5 } , - \frac { 4 } { 5 } \right)$ 
H) $\langle 6 , - 8 \rangle$

Question 16

Describe the surface whose equation in cylindrical coordinates is $z = r ^ { 2 }$ .

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 17

Find the component form of a + b given that a and b are unit vectors.

Accepted Answer

The answer of Find the component form of a +...

Question 18

Let f(x, y) = 5
(a) Evaluate $f ( - 1 , - 1 )$ .(b) Find the domain of f.(c) Find the range of f.

Accepted Answer

(a) 5
(b)

Question 19

Use vectors to determine whether the points P(0, 0, 0), Q(5, 0, 0), R(2, 6, 6), and S(7, 6, 6) are coplanar.

Accepted Answer

The answer of Use vectors to determine whether the points...

Question 20

Find the distance from the point (-1, 2, 3)to the plane x + 2y + 2z = -6.

Accepted Answer

A) 1 
B) 2 
C) 6 
D)  $\frac { 5 } { 3 }$ 
E) 15
F) $\frac { 3 } { 5 }$ 
G)4

H)5 
A) 1 
B) 2 
C) 6 
D)  $\frac { 5 } { 3 }$ 
E) 15
F) $\frac { 3 } { 5 }$ 
G)4

H)5

If $p=(1,1, \sqrt{2})$ in rectangular coordinates, find the spherical coordinates of P.

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

A person pushing a lawnmower exerts a force of 30 pounds in the direction of the handle, which makes an angle of 30° with the ground. How much work is done in moving the lawnmower 20 feet?

Find an equation of the plane containing P(-1, 2, 1) and the line $\frac { x + 1 } { 2 } = \frac { y } { 5 } = \frac { z - 3 } { 1 }$ .

Let $\mathbf { a } = \langle 5 , - 1 , - 2 \}$ and $\mathbf { b } = \{ 1 , - 4,3 \}$ .(a) Compute $a + 2 b$ .(b) Find $| \mathbf { a } |$ .(c) Find the unit vector in the direction of b.

Given a = $\langle 7,4 \rangle$ , b = $\langle 1,2 \rangle$ , find the vector projection of b onto a.

Determine all values for a such that the vectors $\mathbf { x } = 2 \mathbf { i } + \mathbf { j } + 2 \mathbf { k }$ and $\mathbf { y } = \mathbf { i } + 2 \mathbf { j } + a \mathbf { k }$ will form a 60° angle.

If vector a = 3i + 4j - 2k; and vector b = 2i - j + 5k, find a vector perpendicular to both vectors.

Find a vector of length 6 with the same direction as $\mathbf { a } = \langle 2,2 , - 1 \rangle$

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

Identify the trace of the surface $x ^ { 2 } = y ^ { 2 } + z ^ { 2 }$ in the plane z = 1.

For vectors a and b, given that $| \mathbf { a } \times \mathbf { b } | = | \mathbf { a } | | \mathbf { b } |$ , which of the following statements is true?

Sketch the graph of $z = x ^ { 2 } + y ^ { 2 }$ in $\mathbb { R } ^ { 3 }$ , and name the surface. $Sketch the graph of z = x ^ { 2 } + y ^ { 2 } in \mathbb { R } ^ { 3 } , and name the surface.$

Describe the vertical traces x = 0 and the horizontal traces z = -1 (if any) for the surfaces $z = x ^ { 2 } + y ^ { 2 }$ and $z ^ { 2 } = x ^ { 2 } + y ^ { 2 }$ .

Find a unit vector that has the same direction as the vector $\langle6 , - 8 \rangle$ .

Describe the surface whose equation in cylindrical coordinates is $z = r ^ { 2 }$ .

Find the component form of a + b given that a and b are unit vectors.

Let f(x, y) = 5 (a) Evaluate $f ( - 1 , - 1 )$ .(b) Find the domain of f.(c) Find the range of f.

Use vectors to determine whether the points P(0, 0, 0), Q(5, 0, 0), R(2, 6, 6), and S(7, 6, 6) are coplanar.

Find the distance from the point (-1, 2, 3)to the plane x + 2y + 2z = -6.

Functions and Models

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Exam 9: Vectors and the Geometry of Space

If p=(1,1,2) p=(1,1, \sqrt{2}) p=(1,1,2​) in rectangular coordinates, find the spherical coordinates of P.

Describe the surface whose equation in cylindrical coordinates is ϕ=3\phi = 3ϕ=3 .

A person pushing a lawnmower exerts a force of 30 pounds in the direction of the handle, which makes an angle of 30° with the ground. How much work is done in moving the lawnmower 20 feet?

Find an equation of the plane containing P(-1, 2, 1) and the line x+12=y5=z−31\frac { x + 1 } { 2 } = \frac { y } { 5 } = \frac { z - 3 } { 1 }2x+1​=5y​=1z−3​ .

Let a=⟨5,−1,−2}\mathbf { a } = \langle 5 , - 1 , - 2 \}a=⟨5,−1,−2} and b={1,−4,3}\mathbf { b } = \{ 1 , - 4,3 \}b={1,−4,3} .(a) Compute a+2ba + 2 ba+2b .(b) Find ∣a∣| \mathbf { a } |∣a∣ .(c) Find the unit vector in the direction of b.

Given a = ⟨7,4⟩\langle 7,4 \rangle⟨7,4⟩ , b = ⟨1,2⟩\langle 1,2 \rangle⟨1,2⟩ , find the vector projection of b onto a.

Determine all values for a such that the vectors x=2i+j+2k\mathbf { x } = 2 \mathbf { i } + \mathbf { j } + 2 \mathbf { k }x=2i+j+2k and y=i+2j+ak\mathbf { y } = \mathbf { i } + 2 \mathbf { j } + a \mathbf { k }y=i+2j+ak will form a 60° angle.

If vector a = 3i + 4j - 2k; and vector b = 2i - j + 5k, find a vector perpendicular to both vectors.

Find a vector of length 6 with the same direction as a=⟨2,2,−1⟩\mathbf { a } = \langle 2,2 , - 1 \ranglea=⟨2,2,−1⟩

Find the domain of the function f(x, y) = ex−y2e ^ { x - y ^ { 2 } }ex−y2 .

Identify the trace of the surface x2=y2+z2x ^ { 2 } = y ^ { 2 } + z ^ { 2 }x2=y2+z2 in the plane z = 1.

For vectors a and b, given that ∣a×b∣=∣a∣∣b∣| \mathbf { a } \times \mathbf { b } | = | \mathbf { a } | | \mathbf { b } |∣a×b∣=∣a∣∣b∣ , which of the following statements is true?

Sketch the graph of z=x2+y2z = x ^ { 2 } + y ^ { 2 }z=x2+y2 in R3\mathbb { R } ^ { 3 }R3 , and name the surface.

Describe the vertical traces x = 0 and the horizontal traces z = -1 (if any) for the surfaces z=x2+y2z = x ^ { 2 } + y ^ { 2 }z=x2+y2 and z2=x2+y2z ^ { 2 } = x ^ { 2 } + y ^ { 2 }z2=x2+y2 .

Find a unit vector that has the same direction as the vector ⟨6,−8⟩\langle6 , - 8 \rangle⟨6,−8⟩ .

Describe the surface whose equation in cylindrical coordinates is z=r2z = r ^ { 2 }z=r2 .

Find the component form of a + b given that a and b are unit vectors.

Let f(x, y) = 5 (a) Evaluate f(−1,−1)f ( - 1 , - 1 )f(−1,−1) .(b) Find the domain of f.(c) Find the range of f.

Use vectors to determine whether the points P(0, 0, 0), Q(5, 0, 0), R(2, 6, 6), and S(7, 6, 6) are coplanar.

Find the distance from the point (-1, 2, 3)to the plane x + 2y + 2z = -6.

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

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Differential Equations

Infinite Sequences and Series

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Filters

If $p=(1,1, \sqrt{2})$ in rectangular coordinates, find the spherical coordinates of P.

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

Find an equation of the plane containing P(-1, 2, 1) and the line $\frac { x + 1 } { 2 } = \frac { y } { 5 } = \frac { z - 3 } { 1 }$ .

Let $\mathbf { a } = \langle 5 , - 1 , - 2 \}$ and $\mathbf { b } = \{ 1 , - 4,3 \}$ .(a) Compute $a + 2 b$ .(b) Find $| \mathbf { a } |$ .(c) Find the unit vector in the direction of b.

Given a = $\langle 7,4 \rangle$ , b = $\langle 1,2 \rangle$ , find the vector projection of b onto a.

Determine all values for a such that the vectors $\mathbf { x } = 2 \mathbf { i } + \mathbf { j } + 2 \mathbf { k }$ and $\mathbf { y } = \mathbf { i } + 2 \mathbf { j } + a \mathbf { k }$ will form a 60° angle.

Find a vector of length 6 with the same direction as $\mathbf { a } = \langle 2,2 , - 1 \rangle$

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

Identify the trace of the surface $x ^ { 2 } = y ^ { 2 } + z ^ { 2 }$ in the plane z = 1.

For vectors a and b, given that $| \mathbf { a } \times \mathbf { b } | = | \mathbf { a } | | \mathbf { b } |$ , which of the following statements is true?

Sketch the graph of $z = x ^ { 2 } + y ^ { 2 }$ in $\mathbb { R } ^ { 3 }$ , and name the surface. $Sketch the graph of z = x ^ { 2 } + y ^ { 2 } in \mathbb { R } ^ { 3 } , and name the surface.$

Describe the vertical traces x = 0 and the horizontal traces z = -1 (if any) for the surfaces $z = x ^ { 2 } + y ^ { 2 }$ and $z ^ { 2 } = x ^ { 2 } + y ^ { 2 }$ .

Find a unit vector that has the same direction as the vector $\langle6 , - 8 \rangle$ .

Describe the surface whose equation in cylindrical coordinates is $z = r ^ { 2 }$ .

Let f(x, y) = 5 (a) Evaluate $f ( - 1 , - 1 )$ .(b) Find the domain of f.(c) Find the range of f.