Question 1

Find the set of intersection of the surfaces whose equations in spherical coordinates are $\rho = 3$ and $\phi = 0$ .

Accepted Answer

A point on the z-axi

Question 2

Find the volume of the parallelepiped below given P = (1, -3, 2), Q = (3, -1, 3), R = (2, 1, -4), and S = (-1, 2, 1).

Accepted Answer

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

Question 3

The planes P1: x + 2y + 3z = 2 and P2: -2x + 3y + 2z = -4 both contain the point P (2, 0, 0).Find a vector equation r = OP0 + td for the line of intersection of these planes.

Accepted Answer

Question 4

Find parametric equations and symmetric equations of the line passing through the points A and B shown below.

Accepted Answer

x = t, y =

Question 5

Determine whether the points A $( 1,3,5 )$ , B $( 8,4,2 )$ , and C $( - 13,1,11 )$ lie on a straight line.

Accepted Answer

The answer of Determine whether the points A \(( 1,3,5...

Question 6

Are the planes 3x - y + 5z = 13 and x + 7y - 2z = 4 perpendicular to each other?

Accepted Answer

The answer of Are the planes 3x - y +...

Question 7

Convert the point $( 1 , - \sqrt { 3 } , - 2 )$ to cylindrical and spherical coordinates.

Accepted Answer

Cylindrical coordina

Question 8

Let f(x, y) = $x e ^ { y / x }$ (a) Evaluate $f ( 2,0 )$ .(b) Find the domain of f.(c) Find the range of f.

Accepted Answer

(a) 2
(b)

Question 9

Find the point of intersection of the line x = 2t, y = t - 1, z = 3t + 4 and the plane x - 5y + 3z = 47.

Accepted Answer

A) (10, 4, 19) 
B) (-10, 4, 19) 
C) (0, -4, -1) 
D) (10, -4, 19) 
E) (0, -1, 4)
F)(0, 1, -4)
G)(0, -1, -4)
H)(10, -4, -19) 
A) (10, 4, 19) 
B) (-10, 4, 19) 
C) (0, -4, -1) 
D) (10, -4, 19) 
E) (0, -1, 4)
F)(0, 1, -4)
G)(0, -1, -4)
H)(10, -4, -19)

Question 10

Find an equation of the set of all points P such that the distance from P to A(0, 0, 0) is twice the distance from P to B(0, 0, 1). Describe the set.

Accepted Answer

@#IMG-DLM& , a spher

Question 11

Given a quadric surface $x ^ { 2 } + 2 y ^ { 2 } + z = 0$ .(a) Identify and sketch the surface.(b) Find the coordinates of the point(s) of intersection of the line passing through the points $( 1,1,0 )$ and $( 0,0 , - 4 )$ and the surface.

Accepted Answer

(a) Ellipt

Question 12

Find the dot product of the vectors $\langle 1,2,3 \rangle$ and $\langle3,0 , - 7 \rangle$ .

Accepted Answer

A) -24 
B) -21 
C) -18 
D) -16 
E) 24
F)21
G)18
H)16 
A) -24 
B) -21 
C) -18 
D) -16 
E) 24
F)21
G)18
H)16

Question 13

Find an equation of the plane through the point P = (2, 1, -4) and perpendicular to the line x = 2 + 3t, y = 1 - 4t, z = 3 + 3t.

Accepted Answer

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

Question 14

Find the value d for which the plane x - y + 2z = d passes through the point (1, 2, 3).

Accepted Answer

A) 2 
B) 3 
C) 4 
D) 5 
E) 6
F)7
G)8
H)9 
A) 2 
B) 3 
C) 4 
D) 5 
E) 6
F)7
G)8
H)9

Question 15

Find a unit vector that has the same direction as the vector $\langle 2,2,1 \rangle$ .

Accepted Answer

A)  $\langle 1,1,1 \rangle$ 
B)  $\langle 1,0,0 \rangle$ 
C)  $\langle 0,1,0 \rangle$ 
D)  $\langle 0,0,1 \rangle$ 
E)  $\langle 1,1,0 \rangle$ 
F) $\left\langle \frac { 2 } { 3 } , \frac { 2 } { 3 } , \frac { 1 } { 3 } \right\rangle$ 
G) $\left\langle \frac { 1 } { 2 } , \frac { 1 } { 2 } , 1 \right\rangle$ 
H) $\left\langle \frac { 1 } { 3 } , \frac { 1 } { 3 } , 1 \right\rangle$ 
A)  $\langle 1,1,1 \rangle$ 
B)  $\langle 1,0,0 \rangle$ 
C)  $\langle 0,1,0 \rangle$ 
D)  $\langle 0,0,1 \rangle$ 
E)  $\langle 1,1,0 \rangle$ 
F) $\left\langle \frac { 2 } { 3 } , \frac { 2 } { 3 } , \frac { 1 } { 3 } \right\rangle$ 
G) $\left\langle \frac { 1 } { 2 } , \frac { 1 } { 2 } , 1 \right\rangle$ 
H) $\left\langle \frac { 1 } { 3 } , \frac { 1 } { 3 } , 1 \right\rangle$

Question 16

Describe the surface whose equation in cylindrical coordinates is $z = 4 r$ .

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 cosine of the acute angle between the two diagonals of a rectangle with length 3 and width 2.

Accepted Answer

A)  $- \frac { 5 } { 13 }$ 
B) 5 
C)  $\frac { 2 } { 13 }$ 
D)  $\frac { \sqrt { 2 } } { 2 }$ 
E)  $\frac { 5 } { 13 }$ 
F) $\frac { 5 \sqrt { 13 } } { 13 }$ 
G) $\frac { 2 } { 5 }$ 
H) $\frac { \sqrt { 3 } } { 5 }$ 
A)  $- \frac { 5 } { 13 }$ 
B) 5 
C)  $\frac { 2 } { 13 }$ 
D)  $\frac { \sqrt { 2 } } { 2 }$ 
E)  $\frac { 5 } { 13 }$ 
F) $\frac { 5 \sqrt { 13 } } { 13 }$ 
G) $\frac { 2 } { 5 }$ 
H) $\frac { \sqrt { 3 } } { 5 }$

Question 18

Find $| \mathbf { a } |$ , a + b, $\mathbf { a } - \mathbf { b }$ , 2a, and 3a + 4b given $\mathrm { a } = \langle - 1,2 \rangle$ and $b = \langle 4,3 \rangle$ .

Accepted Answer

@#IMG-DLM& = @#IMG-DLM& , a +

Question 19

Given a triangle with vertices A(1, 2), B(2, 3), and C(5, 2), find the angle $\angle A B C$ correct to the nearest degree.

Accepted Answer

The answer of Given a triangle with vertices A(1, 2),...

Question 20

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 $\mathbf { a } \cdot \mathbf { d } - \mathbf { b } \cdot \mathbf { c }$ .

Accepted Answer

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

Find the set of intersection of the surfaces whose equations in spherical coordinates are $\rho = 3$ and $\phi = 0$ .

Find the volume of the parallelepiped below given P = (1, -3, 2), Q = (3, -1, 3), R = (2, 1, -4), and S = (-1, 2, 1).

The planes P₁: x + 2y + 3z = 2 and P₂: -2x + 3y + 2z = -4 both contain the point P (2, 0, 0).Find a vector equation r = OP₀ + td for the line of intersection of these planes.

Find parametric equations and symmetric equations of the line passing through the points A and B shown below.

Determine whether the points A $( 1,3,5 )$ , B $( 8,4,2 )$ , and C $( - 13,1,11 )$ lie on a straight line.

Are the planes 3x - y + 5z = 13 and x + 7y - 2z = 4 perpendicular to each other?

Convert the point $( 1 , - \sqrt { 3 } , - 2 )$ to cylindrical and spherical coordinates.

Let f(x, y) = $x e ^ { y / x }$ (a) Evaluate $f ( 2,0 )$ .(b) Find the domain of f.(c) Find the range of f.

Find the point of intersection of the line x = 2t, y = t - 1, z = 3t + 4 and the plane x - 5y + 3z = 47.

Find an equation of the set of all points P such that the distance from P to A(0, 0, 0) is twice the distance from P to B(0, 0, 1). Describe the set.

Given a quadric surface $x ^ { 2 } + 2 y ^ { 2 } + z = 0$ .(a) Identify and sketch the surface.(b) Find the coordinates of the point(s) of intersection of the line passing through the points $( 1,1,0 )$ and $( 0,0 , - 4 )$ and the surface.

Find the dot product of the vectors $\langle 1,2,3 \rangle$ and $\langle3,0 , - 7 \rangle$ .

Find an equation of the plane through the point P = (2, 1, -4) and perpendicular to the line x = 2 + 3t, y = 1 - 4t, z = 3 + 3t.

Find the value d for which the plane x - y + 2z = d passes through the point (1, 2, 3).

Find a unit vector that has the same direction as the vector $\langle 2,2,1 \rangle$ .

Describe the surface whose equation in cylindrical coordinates is $z = 4 r$ .

Find the cosine of the acute angle between the two diagonals of a rectangle with length 3 and width 2.

Find $| \mathbf { a } |$ , a + b, $\mathbf { a } - \mathbf { b }$ , 2a, and 3a + 4b given $\mathrm { a } = \langle - 1,2 \rangle$ and $b = \langle 4,3 \rangle$ .

Given a triangle with vertices A(1, 2), B(2, 3), and C(5, 2), find the angle $\angle A B C$ correct to the nearest degree.

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 $\mathbf { a } \cdot \mathbf { d } - \mathbf { b } \cdot \mathbf { c }$ .

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

Exam 9: Vectors and the Geometry of Space

Find the set of intersection of the surfaces whose equations in spherical coordinates are ρ=3\rho = 3ρ=3 and ϕ=0\phi = 0ϕ=0 .

Find the volume of the parallelepiped below given P = (1, -3, 2), Q = (3, -1, 3), R = (2, 1, -4), and S = (-1, 2, 1).

The planes P1: x + 2y + 3z = 2 and P2: -2x + 3y + 2z = -4 both contain the point P (2, 0, 0).Find a vector equation r = OP0 + td for the line of intersection of these planes.

Find parametric equations and symmetric equations of the line passing through the points A and B shown below.

Determine whether the points A (1,3,5)( 1,3,5 )(1,3,5) , B (8,4,2)( 8,4,2 )(8,4,2) , and C (−13,1,11)( - 13,1,11 )(−13,1,11) lie on a straight line.

Are the planes 3x - y + 5z = 13 and x + 7y - 2z = 4 perpendicular to each other?

Convert the point (1,−3,−2)( 1 , - \sqrt { 3 } , - 2 )(1,−3​,−2) to cylindrical and spherical coordinates.

Let f(x, y) = xey/xx e ^ { y / x }xey/x (a) Evaluate f(2,0)f ( 2,0 )f(2,0) .(b) Find the domain of f.(c) Find the range of f.

Find the point of intersection of the line x = 2t, y = t - 1, z = 3t + 4 and the plane x - 5y + 3z = 47.

Find an equation of the set of all points P such that the distance from P to A(0, 0, 0) is twice the distance from P to B(0, 0, 1). Describe the set.

Given a quadric surface x2+2y2+z=0x ^ { 2 } + 2 y ^ { 2 } + z = 0x2+2y2+z=0 .(a) Identify and sketch the surface.(b) Find the coordinates of the point(s) of intersection of the line passing through the points (1,1,0)( 1,1,0 )(1,1,0) and (0,0,−4)( 0,0 , - 4 )(0,0,−4) and the surface.

Find the dot product of the vectors ⟨1,2,3⟩\langle 1,2,3 \rangle⟨1,2,3⟩ and ⟨3,0,−7⟩\langle3,0 , - 7 \rangle⟨3,0,−7⟩ .

Find an equation of the plane through the point P = (2, 1, -4) and perpendicular to the line x = 2 + 3t, y = 1 - 4t, z = 3 + 3t.

Find the value d for which the plane x - y + 2z = d passes through the point (1, 2, 3).

Find a unit vector that has the same direction as the vector ⟨2,2,1⟩\langle 2,2,1 \rangle⟨2,2,1⟩ .

Describe the surface whose equation in cylindrical coordinates is z=4rz = 4 rz=4r .

Find the cosine of the acute angle between the two diagonals of a rectangle with length 3 and width 2.

Find ∣a∣| \mathbf { a } |∣a∣ , a + b, a−b\mathbf { a } - \mathbf { b }a−b , 2a, and 3a + 4b given a=⟨−1,2⟩\mathrm { a } = \langle - 1,2 \ranglea=⟨−1,2⟩ and b=⟨4,3⟩b = \langle 4,3 \rangleb=⟨4,3⟩ .

Given a triangle with vertices A(1, 2), B(2, 3), and C(5, 2), find the angle ∠ABC\angle A B C∠ABC correct to the nearest degree.

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

Find the set of intersection of the surfaces whose equations in spherical coordinates are $\rho = 3$ and $\phi = 0$ .

The planes P₁: x + 2y + 3z = 2 and P₂: -2x + 3y + 2z = -4 both contain the point P (2, 0, 0).Find a vector equation r = OP₀ + td for the line of intersection of these planes.

Determine whether the points A $( 1,3,5 )$ , B $( 8,4,2 )$ , and C $( - 13,1,11 )$ lie on a straight line.

Convert the point $( 1 , - \sqrt { 3 } , - 2 )$ to cylindrical and spherical coordinates.

Let f(x, y) = $x e ^ { y / x }$ (a) Evaluate $f ( 2,0 )$ .(b) Find the domain of f.(c) Find the range of f.

Given a quadric surface $x ^ { 2 } + 2 y ^ { 2 } + z = 0$ .(a) Identify and sketch the surface.(b) Find the coordinates of the point(s) of intersection of the line passing through the points $( 1,1,0 )$ and $( 0,0 , - 4 )$ and the surface.

Find the dot product of the vectors $\langle 1,2,3 \rangle$ and $\langle3,0 , - 7 \rangle$ .

Find a unit vector that has the same direction as the vector $\langle 2,2,1 \rangle$ .

Describe the surface whose equation in cylindrical coordinates is $z = 4 r$ .

Find $| \mathbf { a } |$ , a + b, $\mathbf { a } - \mathbf { b }$ , 2a, and 3a + 4b given $\mathrm { a } = \langle - 1,2 \rangle$ and $b = \langle 4,3 \rangle$ .

Given a triangle with vertices A(1, 2), B(2, 3), and C(5, 2), find the angle $\angle A B C$ correct to the nearest degree.