Question 1

Find the point at which the following three planes intersect:
x - 2y + z = 5
2x - y + z = 1
-2x + y + z = 3

Accepted Answer

The answer of Find the point at which the following...

Question 2

Find the distance between the following two lines.x = -t
x = 3 + t
y = t
and
y = 3t
z = 2t
z = 5 - 4t

Accepted Answer

The answer of Find the distance between the following two...

Question 3

Find the set of intersection of the surfaces whose equations in spherical coordinates are $\rho = 4$ and $\theta = \frac { \pi } { 6 }$ .

Accepted Answer

A circle r

Question 4

Find a unit vector that has the same direction as 2i - 4j + 7k.

Accepted Answer

The answer of Find a unit vector that has the...

Question 5

Sketch the solid given in spherical coordinates by $0 \leq \theta \leq \pi , 0 \leq \phi \leq \frac { \pi } { 4 } , \rho \leq 4 \sec \phi$ .

Accepted Answer

The answer of Sketch the solid given in spherical coordinates...

Question 6

A constant force with vector representation F = i + 2j moves an object along a straight line from the point (2, 4) to the point (5, 7). Find the work done in foot-pounds if force is measured in pounds and distance is measured in feet.

Accepted Answer

A) 6 
B) 7 
C) 8 
D) 9 
E) 10
F)11
G)12
H)13 
A) 6 
B) 7 
C) 8 
D) 9 
E) 10
F)11
G)12
H)13

Question 7

Let a = 3i - 2j + k, b = i -3j + 5k, and c = 2i + j - 4k. Do these vectors form a right triangle? Show why or why not.

Accepted Answer

Yes, the vectors for

Question 8

Find the dot product of the vectors $\langle 1,2 \rangle$ and $\langle 4,5 \rangle$ .

Accepted Answer

A) 14 
B) 15 
C) 16 
D) 17 
E) 18
F)19
G)20
H)21 
A) 14 
B) 15 
C) 16 
D) 17 
E) 18
F)19
G)20
H)21

Question 9

Let l and $ l^{\prime} $ be two lines in space given by the equations
x = 3 + t
x = -1 + t
$ l $ :
y = 1 - t $ l^{\prime} $ :
y = 2t
z = 2t
z = 1 + kt
Find all values of k (if any) for which l and $l ^ { \prime }$ are perpendicular.

Accepted Answer

The answer of Let l and $ l^{\prime} $ be...

Question 10

Sketch the graph of $\frac { x ^ { 2 } } { 4 } + y ^ { 2 } - z ^ { 2 } = 1$ in $\mathbb { R } ^ { 3 }$ , and name the surface.

Accepted Answer

Elliptic h

Question 11

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

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

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 13

For the given forces $F _ { 1 }$ and $F _ { 2 }$ , compute the magnitude of the resulting force and its direction.

Accepted Answer

@#IMG-DLM& . Directi

Question 14

Find the equation of the sphere, in standard form, one of whose diameters has $( - 5,2,9 )$ and (3, 6, 1) as endpoints.

Accepted Answer

The answer of Find the equation of the sphere, in...

Question 15

Find a vector v for a vector equation $\mathbf { r } = \mathbf { r } _ { 0 } + t \mathbf { v }$ of the line passing through the points (1, 2, 3) and (4, 5, 6).

Accepted Answer

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

Question 16

Sketch the region bounded by the surfaces with equations $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 4 z = 0$ and $z = \sqrt { x ^ { 2 } + y ^ { 2 } }$ .

Accepted Answer

The answer of Sketch the region bounded by the surfaces...

Question 17

Compute $| \mathbf { a } \times \mathbf { b } |$ if $| \mathbf { a } | = 3$ , $| \mathbf { b } | = 7$ , and $\mathbf { a } \cdot \mathbf { b } = \mathbf { 0 }$ .

Accepted Answer

The answer of Compute \(| \mathbf { a } \times...

Question 18

Given the points A = (2, 3, 1), B = (4, -1, 5), and O = (0, 0, 0), find the distance from O to the line through A and B.

Accepted Answer

The answer of Given the points A = (2, 3,...

Question 19

Identify the surface $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 3$ .

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 20

Let a = $\langle - 2,3,1 \rangle$ and b = $\langle 6 , c , - 3 \rangle$ .(a) Find the value of c such that a and b are orthogonal.(b) Find the value of c such that a and b are parallel.

Accepted Answer

(a) c = 5

Find the point at which the following three planes intersect: x - 2y + z = 5 2x - y + z = 1 -2x + y + z = 3

Find the distance between the following two lines.x = -t x = 3 + t y = t and y = 3t z = 2t z = 5 - 4t

Find the set of intersection of the surfaces whose equations in spherical coordinates are $\rho = 4$ and $\theta = \frac { \pi } { 6 }$ .

Find a unit vector that has the same direction as 2i - 4j + 7k.

Sketch the solid given in spherical coordinates by $0 \leq \theta \leq \pi , 0 \leq \phi \leq \frac { \pi } { 4 } , \rho \leq 4 \sec \phi$ . $Sketch the solid given in spherical coordinates by 0 \leq \theta \leq \pi , 0 \leq \phi \leq \frac { \pi } { 4 } , \rho \leq 4 \sec \phi .$

A constant force with vector representation F = i + 2j moves an object along a straight line from the point (2, 4) to the point (5, 7). Find the work done in foot-pounds if force is measured in pounds and distance is measured in feet.

Let a = 3i - 2j + k, b = i -3j + 5k, and c = 2i + j - 4k. Do these vectors form a right triangle? Show why or why not.

Find the dot product of the vectors $\langle 1,2 \rangle$ and $\langle 4,5 \rangle$ .

Let l and $l^{\prime}$ be two lines in space given by the equations x = 3 + t x = -1 + t $l$ : y = 1 - t $l^{\prime}$ : y = 2t z = 2t z = 1 + kt Find all values of k (if any) for which l and $l ^ { \prime }$ are perpendicular.

Sketch the graph of $\frac { x ^ { 2 } } { 4 } + y ^ { 2 } - z ^ { 2 } = 1$ in $\mathbb { R } ^ { 3 }$ , and name the surface. $Sketch the graph of \frac { x ^ { 2 } } { 4 } + y ^ { 2 } - z ^ { 2 } = 1 in \mathbb { R } ^ { 3 } , and name the surface.$

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

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

For the given forces $F _ { 1 }$ and $F _ { 2 }$ , compute the magnitude of the resulting force and its direction. $For the given forces F _ { 1 } and F _ { 2 } , compute the magnitude of the resulting force and its direction.$

Find the equation of the sphere, in standard form, one of whose diameters has $( - 5,2,9 )$ and (3, 6, 1) as endpoints.

Find a vector v for a vector equation $\mathbf { r } = \mathbf { r } _ { 0 } + t \mathbf { v }$ of the line passing through the points (1, 2, 3) and (4, 5, 6).

Sketch the region bounded by the surfaces with equations $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 4 z = 0$ and $z = \sqrt { x ^ { 2 } + y ^ { 2 } }$ . $Sketch the region bounded by the surfaces with equations x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 4 z = 0 and z = \sqrt { x ^ { 2 } + y ^ { 2 } } .$

Compute $| \mathbf { a } \times \mathbf { b } |$ if $| \mathbf { a } | = 3$ , $| \mathbf { b } | = 7$ , and $\mathbf { a } \cdot \mathbf { b } = \mathbf { 0 }$ .

Given the points A = (2, 3, 1), B = (4, -1, 5), and O = (0, 0, 0), find the distance from O to the line through A and B.

Identify the surface $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 3$ .

Let a = $\langle - 2,3,1 \rangle$ and b = $\langle 6 , c , - 3 \rangle$ .(a) Find the value of c such that a and b are orthogonal.(b) Find the value of c such that a and b are parallel.

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 point at which the following three planes intersect: x - 2y + z = 5 2x - y + z = 1 -2x + y + z = 3

Find the distance between the following two lines.x = -t x = 3 + t y = t and y = 3t z = 2t z = 5 - 4t

Find the set of intersection of the surfaces whose equations in spherical coordinates are ρ=4\rho = 4ρ=4 and θ=π6\theta = \frac { \pi } { 6 }θ=6π​ .

Find a unit vector that has the same direction as 2i - 4j + 7k.

Sketch the solid given in spherical coordinates by 0≤θ≤π,0≤ϕ≤π4,ρ≤4sec⁡ϕ0 \leq \theta \leq \pi , 0 \leq \phi \leq \frac { \pi } { 4 } , \rho \leq 4 \sec \phi0≤θ≤π,0≤ϕ≤4π​,ρ≤4secϕ .

A constant force with vector representation F = i + 2j moves an object along a straight line from the point (2, 4) to the point (5, 7). Find the work done in foot-pounds if force is measured in pounds and distance is measured in feet.

Let a = 3i - 2j + k, b = i -3j + 5k, and c = 2i + j - 4k. Do these vectors form a right triangle? Show why or why not.

Find the dot product of the vectors ⟨1,2⟩\langle 1,2 \rangle⟨1,2⟩ and ⟨4,5⟩\langle 4,5 \rangle⟨4,5⟩ .

Let l and l′ l^{\prime} l′ be two lines in space given by the equations x = 3 + t x = -1 + t l l l : y = 1 - t l′ l^{\prime} l′ : y = 2t z = 2t z = 1 + kt Find all values of k (if any) for which l and l′l ^ { \prime }l′ are perpendicular.

Sketch the graph of x24+y2−z2=1\frac { x ^ { 2 } } { 4 } + y ^ { 2 } - z ^ { 2 } = 14x2​+y2−z2=1 in R3\mathbb { R } ^ { 3 }R3 , and name the surface.

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

Identify the surface 2=y2+z22 = y ^ { 2 } + z ^ { 2 }2=y2+z2 .

For the given forces F1F _ { 1 }F1​ and F2F _ { 2 }F2​ , compute the magnitude of the resulting force and its direction.

Find the equation of the sphere, in standard form, one of whose diameters has (−5,2,9)( - 5,2,9 )(−5,2,9) and (3, 6, 1) as endpoints.

Find a vector v for a vector equation r=r0+tv\mathbf { r } = \mathbf { r } _ { 0 } + t \mathbf { v }r=r0​+tv of the line passing through the points (1, 2, 3) and (4, 5, 6).

Sketch the region bounded by the surfaces with equations x2+y2+z2−4z=0x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 4 z = 0x2+y2+z2−4z=0 and z=x2+y2z = \sqrt { x ^ { 2 } + y ^ { 2 } }z=x2+y2​ .

Compute ∣a×b∣| \mathbf { a } \times \mathbf { b } |∣a×b∣ if ∣a∣=3| \mathbf { a } | = 3∣a∣=3 , ∣b∣=7| \mathbf { b } | = 7∣b∣=7 , and a⋅b=0\mathbf { a } \cdot \mathbf { b } = \mathbf { 0 }a⋅b=0 .

Given the points A = (2, 3, 1), B = (4, -1, 5), and O = (0, 0, 0), find the distance from O to the line through A and B.

Identify the surface x2+y2+z2=3x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 3x2+y2+z2=3 .

Let a = ⟨−2,3,1⟩\langle - 2,3,1 \rangle⟨−2,3,1⟩ and b = ⟨6,c,−3⟩\langle 6 , c , - 3 \rangle⟨6,c,−3⟩ .(a) Find the value of c such that a and b are orthogonal.(b) Find the value of c such that a and b are parallel.

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 $\rho = 4$ and $\theta = \frac { \pi } { 6 }$ .

Sketch the solid given in spherical coordinates by $0 \leq \theta \leq \pi , 0 \leq \phi \leq \frac { \pi } { 4 } , \rho \leq 4 \sec \phi$ . $Sketch the solid given in spherical coordinates by 0 \leq \theta \leq \pi , 0 \leq \phi \leq \frac { \pi } { 4 } , \rho \leq 4 \sec \phi .$

Find the dot product of the vectors $\langle 1,2 \rangle$ and $\langle 4,5 \rangle$ .

Let l and $l^{\prime}$ be two lines in space given by the equations x = 3 + t x = -1 + t $l$ : y = 1 - t $l^{\prime}$ : y = 2t z = 2t z = 1 + kt Find all values of k (if any) for which l and $l ^ { \prime }$ are perpendicular.

Sketch the graph of $\frac { x ^ { 2 } } { 4 } + y ^ { 2 } - z ^ { 2 } = 1$ in $\mathbb { R } ^ { 3 }$ , and name the surface. $Sketch the graph of \frac { x ^ { 2 } } { 4 } + y ^ { 2 } - z ^ { 2 } = 1 in \mathbb { R } ^ { 3 } , and name the surface.$

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

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

For the given forces $F _ { 1 }$ and $F _ { 2 }$ , compute the magnitude of the resulting force and its direction. $For the given forces F _ { 1 } and F _ { 2 } , compute the magnitude of the resulting force and its direction.$

Find the equation of the sphere, in standard form, one of whose diameters has $( - 5,2,9 )$ and (3, 6, 1) as endpoints.

Find a vector v for a vector equation $\mathbf { r } = \mathbf { r } _ { 0 } + t \mathbf { v }$ of the line passing through the points (1, 2, 3) and (4, 5, 6).

Sketch the region bounded by the surfaces with equations $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 4 z = 0$ and $z = \sqrt { x ^ { 2 } + y ^ { 2 } }$ . $Sketch the region bounded by the surfaces with equations x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 4 z = 0 and z = \sqrt { x ^ { 2 } + y ^ { 2 } } .$

Compute $| \mathbf { a } \times \mathbf { b } |$ if $| \mathbf { a } | = 3$ , $| \mathbf { b } | = 7$ , and $\mathbf { a } \cdot \mathbf { b } = \mathbf { 0 }$ .

Identify the surface $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 3$ .

Let a = $\langle - 2,3,1 \rangle$ and b = $\langle 6 , c , - 3 \rangle$ .(a) Find the value of c such that a and b are orthogonal.(b) Find the value of c such that a and b are parallel.