Question 1

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

Accepted Answer

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

Question 2

Convert $\left( 1 , \frac { \pi } { 4 } , \frac { \pi } { 4 } \right)$ from spherical coordinates to rectangular coordinates.

Accepted Answer

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

Question 3

Find the intersection point of the line $\mathbf { r } = \langle 1,0,2 \rangle + \langle 2 , - 2,1 \rangle t$ and the plane 3x + 4y + 6z = 7.

Accepted Answer

(-3, 4, 0)

Question 4

Find $\mathbf { a } \times \mathbf { b }$ , where a and b are given in the figure.

Accepted Answer

The answer of Find \(\mathbf { a } \times \mathbf...

Question 5

Convert $( 1,1 , \sqrt { 2 } )$ from rectangular coordinates to spherical coordinates.

Accepted Answer

A)  $\left( \sqrt { 2 } , \frac { \pi } { 4 } , \frac { \pi } { 4 } \right)$ 
B)  $\left( 2 , \frac { \pi } { 4 } , \frac { \pi } { 4 } \right)$ 
C)  $\left( \sqrt { 2 } , \frac { \pi } { 2 } , \frac { \pi } { 4 } \right)$ 
D)  $\left( 2 , \frac { \pi } { 2 } , \frac { \pi } { 4 } \right)$ 
E)  $\left( \sqrt { 2 } , \frac { \pi } { 4 } , \frac { \pi } { 2 } \right)$ 
F) $\left( 2 , \frac { \pi } { 4 } , \frac { \pi } { 2 } \right)$ 
G) $\left( \sqrt { 2 } , \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$ 
H) $\left( 2 , \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$ 
A)  $\left( \sqrt { 2 } , \frac { \pi } { 4 } , \frac { \pi } { 4 } \right)$ 
B)  $\left( 2 , \frac { \pi } { 4 } , \frac { \pi } { 4 } \right)$ 
C)  $\left( \sqrt { 2 } , \frac { \pi } { 2 } , \frac { \pi } { 4 } \right)$ 
D)  $\left( 2 , \frac { \pi } { 2 } , \frac { \pi } { 4 } \right)$ 
E)  $\left( \sqrt { 2 } , \frac { \pi } { 4 } , \frac { \pi } { 2 } \right)$ 
F) $\left( 2 , \frac { \pi } { 4 } , \frac { \pi } { 2 } \right)$ 
G) $\left( \sqrt { 2 } , \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$ 
H) $\left( 2 , \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$

Question 6

What region of $\mathbb { R } ^ { 3 }$ is represented by the inequality $x ^ { 2 } + y ^ { 2 } \leq 1$ ?

Accepted Answer

A) points inside a circular cylinder with axis the x-axis 
B) points inside a circular cylinder with axis the y-axis 
C) points inside a circular cylinder with axis the z-axis 
D) points inside a sphere of radius 1 
E) points inside a sphere of radius $\sqrt { 2 }$ 
F)points inside a sphere of radius 2
G)points inside a sphere of radius $\sqrt { 3 }$ 
H)points inside a sphere of radius 3 
A) points inside a circular cylinder with axis the x-axis 
B) points inside a circular cylinder with axis the y-axis 
C) points inside a circular cylinder with axis the z-axis 
D) points inside a sphere of radius 1 
E) points inside a sphere of radius $\sqrt { 2 }$ 
F)points inside a sphere of radius 2
G)points inside a sphere of radius $\sqrt { 3 }$ 
H)points inside a sphere of radius 3

Question 7

Find the cross product $\mathbf { a } \times \mathbf { b }$ , where a = i + 2j + 3k and b = 2i - j + k.

Accepted Answer

The answer of Find the cross product \(\mathbf { a...

Question 8

Convert $( 1,1,1 )$ from rectangular coordinates to cylindrical coordinates.

Accepted Answer

A)  $\left( \sqrt { 2 } , \frac { \pi } { 2 } , 1 \right)$ 
B)  $\left( \sqrt { 2 } , \frac { \pi } { 4 } , 1 \right)$ 
C)  $\left( 1 , \frac { \pi } { 2 } , 1 \right)$ 
D)  $\left( 1 , \frac { \pi } { 4 } , 1 \right)$ 
E)  $\left( 1 , \frac { \pi } { 2 } , \sqrt { 2 } \right)$ 
F) $\left( 1 , \frac { \pi } { 4 } , \sqrt { 2 } \right)$ 
G) $\left( 1 , \frac { \pi } { 2 } , 2 \right)$ 
H) $\left( 1 , \frac { \pi } { 4 } , 2 \right)$ 
A)  $\left( \sqrt { 2 } , \frac { \pi } { 2 } , 1 \right)$ 
B)  $\left( \sqrt { 2 } , \frac { \pi } { 4 } , 1 \right)$ 
C)  $\left( 1 , \frac { \pi } { 2 } , 1 \right)$ 
D)  $\left( 1 , \frac { \pi } { 4 } , 1 \right)$ 
E)  $\left( 1 , \frac { \pi } { 2 } , \sqrt { 2 } \right)$ 
F) $\left( 1 , \frac { \pi } { 4 } , \sqrt { 2 } \right)$ 
G) $\left( 1 , \frac { \pi } { 2 } , 2 \right)$ 
H) $\left( 1 , \frac { \pi } { 4 } , 2 \right)$

Question 9

Describe in words or sketch the solid represented in cylindrical coordinates by the inequalities $\frac { \pi } { 6 } \leq \theta \leq \frac { \pi } { 3 } , 0 \leq r \leq 2 , - 1 \leq z \leq 1$ .

Accepted Answer

Wedge of a circular

Question 10

Find the components of a vector a whose initial point is $( 1,1 )$ and terminal point is $( 4 , - 3 )$ .

Accepted Answer

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

Question 11

Find the cosine of the acute angle between the lines x = 4 - 4t, y = 3 - t, z = 1 + 5t and x = 4 - t, y = 3 + 2t, z = 1.

Accepted Answer

The answer of Find the cosine of the acute angle...

Question 12

Find equations of all planes containing the y-axis.

Accepted Answer

ax + cz =

Question 13

Let a and b be vectors. Under what conditions is $\mathbf { a } \times \mathbf { b } = \mathbf { b } \times \mathbf { a }$ ? When is $( 2 \mathbf { a } ) \times ( 3 \mathbf { b } ) = 6 ( \mathbf { a } \times \mathbf { b } )$ ?

Accepted Answer

@#IMG-DLM& when (1) a = b, (2)

Question 14

Convert $( - 1 , \sqrt { 3 } , 2 )$ from rectangular coordinates to spherical coordinates.

Accepted Answer

A)  $\left( 2 , \frac { \pi } { 6 } , \frac { \pi } { 4 } \right)$ 
B)  $\left( 4 , \frac { \pi } { 6 } , \frac { \pi } { 4 } \right)$ 
C)  $\left( \sqrt { 2 } , \frac { \pi } { 6 } , \frac { \pi } { 4 } \right)$ 
D)  $\left( \sqrt { 8 } , \frac { 2 \pi } { 3 } , \frac { \pi } { 4 } \right)$ 
E)  $\left( 2 , \frac { \pi } { 3 } , \frac { \pi } { 4 } \right)$ 
F) $\left( 4 , \frac { \pi } { 3 } , \frac { \pi } { 4 } \right)$ 
G) $\left( \sqrt { 2 } , \frac { 4 \pi } { 3 } , \frac { \pi } { 4 } \right)$ 
H) $\left( \sqrt { 8 } , \frac { \pi } { 3 } , \frac { \pi } { 4 } \right)$ 
A)  $\left( 2 , \frac { \pi } { 6 } , \frac { \pi } { 4 } \right)$ 
B)  $\left( 4 , \frac { \pi } { 6 } , \frac { \pi } { 4 } \right)$ 
C)  $\left( \sqrt { 2 } , \frac { \pi } { 6 } , \frac { \pi } { 4 } \right)$ 
D)  $\left( \sqrt { 8 } , \frac { 2 \pi } { 3 } , \frac { \pi } { 4 } \right)$ 
E)  $\left( 2 , \frac { \pi } { 3 } , \frac { \pi } { 4 } \right)$ 
F) $\left( 4 , \frac { \pi } { 3 } , \frac { \pi } { 4 } \right)$ 
G) $\left( \sqrt { 2 } , \frac { 4 \pi } { 3 } , \frac { \pi } { 4 } \right)$ 
H) $\left( \sqrt { 8 } , \frac { \pi } { 3 } , \frac { \pi } { 4 } \right)$

Question 15

Let f(x, y) = $\ln ( x y )$ (a) Evaluate $f \left( 3 , \frac { 1 } { 3 } \right)$ .(b) Find the domain of f.(c) Find the range of f.

Accepted Answer

(a) 0
(b) x > 0 and

Question 16

If $Q = \left( 1 , \frac { \pi } { 2 } , 3 \right)$ in cylindrical coordinates, find rectangular coordinates of Q.

Accepted Answer

The answer of If \(Q = \left( 1 , \frac...

Question 17

Identify the surface $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ .

Accepted Answer

A) ellipsoid but not a sphere 
B) hyperboloid of one sheet 
C) hyperboloid of two sheets 
D) cylinder 
E) sphere
F)elliptic but not circular paraboloid
G)cone
H)circular paraboloid (figure of revolution) 
A) ellipsoid but not a sphere 
B) hyperboloid of one sheet 
C) hyperboloid of two sheets 
D) cylinder 
E) sphere
F)elliptic but not circular paraboloid
G)cone
H)circular paraboloid (figure of revolution)

Question 18

Find the torque vector $\tau$ of the force F = $\{ 1,1,0 \}$ acting on a rigid body at the point given by the position vector r = $\{ 2,3,0 \}$ .

Accepted Answer

A)  $\{ - 1,0,0 \}$ 
B)  $\{ 0 , - 1,0 \}$ 
C)  $\{ 0,0 , - 1 \}$ 
D)  $\{ - 1,1,0 \}$ 
E)  $\{ 0 , - 1,1 \}$ 
F) $\{ 1,0 , - 1 \}$ 
G) $( - 1,1,1 )$ 
H) $\{ 1 , - 1,1 \}$ 
A)  $\{ - 1,0,0 \}$ 
B)  $\{ 0 , - 1,0 \}$ 
C)  $\{ 0,0 , - 1 \}$ 
D)  $\{ - 1,1,0 \}$ 
E)  $\{ 0 , - 1,1 \}$ 
F) $\{ 1,0 , - 1 \}$ 
G) $( - 1,1,1 )$ 
H) $\{ 1 , - 1,1 \}$

Question 19

Given the vectors u = 2i + j - k and w = i + j + 4k, find a vector of length 2 which is orthogonal to both u and w.

Accepted Answer

The answer of Given the vectors u = 2i +...

Question 20

Let f(x, y) = $\left( x ^ { 2 } + y \right) ^ { 3 }$ . If x = 1, find f(x, 2x).

Accepted Answer

A) 1 
B) 2 
C) 3 
D) 4 
E) 8
F)9
G)16
H)27 
A) 1 
B) 2 
C) 3 
D) 4 
E) 8
F)9
G)16
H)27

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

Convert $\left( 1 , \frac { \pi } { 4 } , \frac { \pi } { 4 } \right)$ from spherical coordinates to rectangular coordinates.

Find the intersection point of the line $\mathbf { r } = \langle 1,0,2 \rangle + \langle 2 , - 2,1 \rangle t$ and the plane 3x + 4y + 6z = 7.

Find $\mathbf { a } \times \mathbf { b }$ , where a and b are given in the figure. $Find \mathbf { a } \times \mathbf { b } , where a and b are given in the figure.$

Convert $( 1,1 , \sqrt { 2 } )$ from rectangular coordinates to spherical coordinates.

What region of $\mathbb { R } ^ { 3 }$ is represented by the inequality $x ^ { 2 } + y ^ { 2 } \leq 1$ ?

Find the cross product $\mathbf { a } \times \mathbf { b }$ , where a = i + 2j + 3k and b = 2i - j + k.

Convert $( 1,1,1 )$ from rectangular coordinates to cylindrical coordinates.

Find the components of a vector a whose initial point is $( 1,1 )$ and terminal point is $( 4 , - 3 )$ .

Find the cosine of the acute angle between the lines x = 4 - 4t, y = 3 - t, z = 1 + 5t and x = 4 - t, y = 3 + 2t, z = 1.

Find equations of all planes containing the y-axis.

Let a and b be vectors. Under what conditions is $\mathbf { a } \times \mathbf { b } = \mathbf { b } \times \mathbf { a }$ ? When is $( 2 \mathbf { a } ) \times ( 3 \mathbf { b } ) = 6 ( \mathbf { a } \times \mathbf { b } )$ ?

Convert $( - 1 , \sqrt { 3 } , 2 )$ from rectangular coordinates to spherical coordinates.

Let f(x, y) = $\ln ( x y )$ (a) Evaluate $f \left( 3 , \frac { 1 } { 3 } \right)$ .(b) Find the domain of f.(c) Find the range of f.

If $Q = \left( 1 , \frac { \pi } { 2 } , 3 \right)$ in cylindrical coordinates, find rectangular coordinates of Q.

Identify the surface $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ .

Find the torque vector $\tau$ of the force F = $\{ 1,1,0 \}$ acting on a rigid body at the point given by the position vector r = $\{ 2,3,0 \}$ .

Given the vectors u = 2i + j - k and w = i + j + 4k, find a vector of length 2 which is orthogonal to both u and w.

Let f(x, y) = $\left( x ^ { 2 } + y \right) ^ { 3 }$ . If x = 1, find f(x, 2x).

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Exam 9: Vectors and the Geometry of Space

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

Convert (1,π4,π4)\left( 1 , \frac { \pi } { 4 } , \frac { \pi } { 4 } \right)(1,4π​,4π​) from spherical coordinates to rectangular coordinates.

Find the intersection point of the line r=⟨1,0,2⟩+⟨2,−2,1⟩t\mathbf { r } = \langle 1,0,2 \rangle + \langle 2 , - 2,1 \rangle tr=⟨1,0,2⟩+⟨2,−2,1⟩t and the plane 3x + 4y + 6z = 7.

Find a×b\mathbf { a } \times \mathbf { b }a×b , where a and b are given in the figure.

Convert (1,1,2)( 1,1 , \sqrt { 2 } )(1,1,2​) from rectangular coordinates to spherical coordinates.

What region of R3\mathbb { R } ^ { 3 }R3 is represented by the inequality x2+y2≤1x ^ { 2 } + y ^ { 2 } \leq 1x2+y2≤1 ?

Find the cross product a×b\mathbf { a } \times \mathbf { b }a×b , where a = i + 2j + 3k and b = 2i - j + k.

Convert (1,1,1)( 1,1,1 )(1,1,1) from rectangular coordinates to cylindrical coordinates.

Describe in words or sketch the solid represented in cylindrical coordinates by the inequalities π6≤θ≤π3,0≤r≤2,−1≤z≤1\frac { \pi } { 6 } \leq \theta \leq \frac { \pi } { 3 } , 0 \leq r \leq 2 , - 1 \leq z \leq 16π​≤θ≤3π​,0≤r≤2,−1≤z≤1 .

Find the components of a vector a whose initial point is (1,1)( 1,1 )(1,1) and terminal point is (4,−3)( 4 , - 3 )(4,−3) .

Find the cosine of the acute angle between the lines x = 4 - 4t, y = 3 - t, z = 1 + 5t and x = 4 - t, y = 3 + 2t, z = 1.

Find equations of all planes containing the y-axis.

Let a and b be vectors. Under what conditions is a×b=b×a\mathbf { a } \times \mathbf { b } = \mathbf { b } \times \mathbf { a }a×b=b×a ? When is (2a)×(3b)=6(a×b)( 2 \mathbf { a } ) \times ( 3 \mathbf { b } ) = 6 ( \mathbf { a } \times \mathbf { b } )(2a)×(3b)=6(a×b) ?

Convert (−1,3,2)( - 1 , \sqrt { 3 } , 2 )(−1,3​,2) from rectangular coordinates to spherical coordinates.

Let f(x, y) = ln⁡(xy)\ln ( x y )ln(xy) (a) Evaluate f(3,13)f \left( 3 , \frac { 1 } { 3 } \right)f(3,31​) .(b) Find the domain of f.(c) Find the range of f.

If Q=(1,π2,3)Q = \left( 1 , \frac { \pi } { 2 } , 3 \right)Q=(1,2π​,3) in cylindrical coordinates, find rectangular coordinates of Q.

Identify the surface x2+y2+z2=4x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4x2+y2+z2=4 .

Find the torque vector τ\tauτ of the force F = {1,1,0}\{ 1,1,0 \}{1,1,0} acting on a rigid body at the point given by the position vector r = {2,3,0}\{ 2,3,0 \}{2,3,0} .

Given the vectors u = 2i + j - k and w = i + j + 4k, find a vector of length 2 which is orthogonal to both u and w.

Let f(x, y) = (x2+y)3\left( x ^ { 2 } + y \right) ^ { 3 }(x2+y)3 . If x = 1, find f(x, 2x).

Functions and Models

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Convert $\left( 1 , \frac { \pi } { 4 } , \frac { \pi } { 4 } \right)$ from spherical coordinates to rectangular coordinates.

Find the intersection point of the line $\mathbf { r } = \langle 1,0,2 \rangle + \langle 2 , - 2,1 \rangle t$ and the plane 3x + 4y + 6z = 7.

Find $\mathbf { a } \times \mathbf { b }$ , where a and b are given in the figure. $Find \mathbf { a } \times \mathbf { b } , where a and b are given in the figure.$

Convert $( 1,1 , \sqrt { 2 } )$ from rectangular coordinates to spherical coordinates.

What region of $\mathbb { R } ^ { 3 }$ is represented by the inequality $x ^ { 2 } + y ^ { 2 } \leq 1$ ?

Find the cross product $\mathbf { a } \times \mathbf { b }$ , where a = i + 2j + 3k and b = 2i - j + k.

Convert $( 1,1,1 )$ from rectangular coordinates to cylindrical coordinates.

Find the components of a vector a whose initial point is $( 1,1 )$ and terminal point is $( 4 , - 3 )$ .

Let a and b be vectors. Under what conditions is $\mathbf { a } \times \mathbf { b } = \mathbf { b } \times \mathbf { a }$ ? When is $( 2 \mathbf { a } ) \times ( 3 \mathbf { b } ) = 6 ( \mathbf { a } \times \mathbf { b } )$ ?

Convert $( - 1 , \sqrt { 3 } , 2 )$ from rectangular coordinates to spherical coordinates.

Let f(x, y) = $\ln ( x y )$ (a) Evaluate $f \left( 3 , \frac { 1 } { 3 } \right)$ .(b) Find the domain of f.(c) Find the range of f.

If $Q = \left( 1 , \frac { \pi } { 2 } , 3 \right)$ in cylindrical coordinates, find rectangular coordinates of Q.

Identify the surface $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ .

Find the torque vector $\tau$ of the force F = $\{ 1,1,0 \}$ acting on a rigid body at the point given by the position vector r = $\{ 2,3,0 \}$ .

Let f(x, y) = $\left( x ^ { 2 } + y \right) ^ { 3 }$ . If x = 1, find f(x, 2x).