Question 1

Given $( - 1,2 )$ and $( 5,3 )$ , find their sum and illustrate geometrically.

Accepted Answer

The answer of Given $( - 1,2 )$ and \((...

Question 2

Find the length of the vector 2i + j - 2k.

Accepted Answer

A)  $\sqrt { 2 }$ 
B)  $\sqrt { 3 }$ 
C) 2 
D)  $\sqrt { 5 }$ 
E)  $\sqrt { 6 }$ 
F) $\sqrt { 7 }$ 
G) $\sqrt { 8 }$ 
H)3 
A)  $\sqrt { 2 }$ 
B)  $\sqrt { 3 }$ 
C) 2 
D)  $\sqrt { 5 }$ 
E)  $\sqrt { 6 }$ 
F) $\sqrt { 7 }$ 
G) $\sqrt { 8 }$ 
H)3

Question 3

Determine whether the lines L1: x = 1 + t, y = 2 + 3t, z = 3 + t and L2: x = 1 + t, y = 3 + 4t, z = 4 + 2t are parallel, intersecting or skew. If they intersect, find the point of intersection.

Accepted Answer

They inter

Question 4

Describe the surface whose equation in cylindrical coordinates is $\rho = 4 \cos \phi$ .

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 5

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

Accepted Answer

A)  $( 0 , \infty )$ 
B)  $[ 0 , \infty )$ 
C)  $( - \infty , \infty )$ 
D)  $( - \infty , 0 )$ 
E)  $( 1 , \infty )$ 
F) $[ 1 , \infty )$ 
G) $( \sqrt { 2 } , \infty )$ 
H) $[ \sqrt { 2 } , \infty )$ 
A)  $( 0 , \infty )$ 
B)  $[ 0 , \infty )$ 
C)  $( - \infty , \infty )$ 
D)  $( - \infty , 0 )$ 
E)  $( 1 , \infty )$ 
F) $[ 1 , \infty )$ 
G) $( \sqrt { 2 } , \infty )$ 
H) $[ \sqrt { 2 } , \infty )$

Question 6

Identify the trace of the surface $x ^ { 2 } = y ^ { 2 } + z ^ { 2 }$ in the plane x = 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 7

Identify the trace of the surface $x = y ^ { 2 } + z ^ { 2 }$ in the plane x = 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 8

Identify the trace of the surface $x = y ^ { 2 } + z ^ { 2 }$ 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 9

Find a unit vector that has the same direction as the vector from the point P $( 1 , - 1,1 )$ to the point Q $( 3,2 , - 5 )$ .

Accepted Answer

A)  $\left( - \frac { 1 } { 2 } , \frac { 1 } { 3 } , - \frac { 1 } { 6 } \right)$ 
B)  $\left\langle \frac { 1 } { 2 } , \frac { 1 } { 3 } , \frac { 1 } { 6 } \right\rangle$ 
C)  $\left\langle \frac { 7 } { 2 } , \frac { 7 } { 3 } , \frac { 7 } { 6 } \right\rangle$ 
D)  $\left\langle \frac { 2 } { 7 } , \frac { 3 } { 7 } , \frac { 6 } { 7 } \right\rangle$ 
E)  $\left\langle \frac { 1 } { 3 } , \frac { 1 } { 4 } , \frac { 1 } { 5 } \right\rangle$ 
F) $\left( \frac { 2 } { 7 } , \frac { 3 } { 7 } , - \frac { 6 } { 7 } \right)$ 
G) $\left( \frac { 2 } { 3 } , - \frac { 3 } { 4 } , \frac { 5 } { 6 } \right)$ 
H) $\left( \frac { 2 } { 3 } , \frac { 3 } { 4 } , - \frac { 5 } { 6 } \right)$ 
A)  $\left( - \frac { 1 } { 2 } , \frac { 1 } { 3 } , - \frac { 1 } { 6 } \right)$ 
B)  $\left\langle \frac { 1 } { 2 } , \frac { 1 } { 3 } , \frac { 1 } { 6 } \right\rangle$ 
C)  $\left\langle \frac { 7 } { 2 } , \frac { 7 } { 3 } , \frac { 7 } { 6 } \right\rangle$ 
D)  $\left\langle \frac { 2 } { 7 } , \frac { 3 } { 7 } , \frac { 6 } { 7 } \right\rangle$ 
E)  $\left\langle \frac { 1 } { 3 } , \frac { 1 } { 4 } , \frac { 1 } { 5 } \right\rangle$ 
F) $\left( \frac { 2 } { 7 } , \frac { 3 } { 7 } , - \frac { 6 } { 7 } \right)$ 
G) $\left( \frac { 2 } { 3 } , - \frac { 3 } { 4 } , \frac { 5 } { 6 } \right)$ 
H) $\left( \frac { 2 } { 3 } , \frac { 3 } { 4 } , - \frac { 5 } { 6 } \right)$

Question 10

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

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 11

Given $(0,3,2 )$ and $( 1,0 , - 3 )$ , find their sum and illustrate geometrically.

Accepted Answer

The answer of Given $(0,3,2 )$ and \(( 1,0 ,...

Question 12

Let a = 3i - j + 2k and b = i - 2j + 3k. Find the scalar projection of a onto b.

Accepted Answer

The answer of Let a = 3i - j +...

Question 13

Find an equation for the plane consisting of all points that are equidistant from the two points (1, 1, 0) and (0, 1, 1).

Accepted Answer

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

Question 14

Find the cross product $\langle 1,2,0\rangle \times \langle 3,4,0 \rangle$ .

Accepted Answer

A)  $\langle0,0 , - 2 \rangle$ 
B)  $\langle 0,0,2 \rangle$ 
C)  $\langle 3,8,0 \rangle$ 
D)  $\langle 8,3,0 \rangle$ 
E)  $\langle 0,3,8 \rangle$ 
F) $\langle 0,8,3 \rangle$ 
G) $\{ 3,0,8 \}$ 
H) $\{ 8,0,3 \}$ 
A)  $\langle0,0 , - 2 \rangle$ 
B)  $\langle 0,0,2 \rangle$ 
C)  $\langle 3,8,0 \rangle$ 
D)  $\langle 8,3,0 \rangle$ 
E)  $\langle 0,3,8 \rangle$ 
F) $\langle 0,8,3 \rangle$ 
G) $\{ 3,0,8 \}$ 
H) $\{ 8,0,3 \}$

Question 15

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

Accepted Answer

A)  $\left( 1 , \frac { \pi } { 3 } , \sqrt { 3 } \right)$ 
B)  $\left( 1 , \frac { \pi } { 6 } , \sqrt { 3 } \right)$ 
C)  $\left( \sqrt { 3 } , \frac { \pi } { 3 } , 1 \right)$ 
D)  $\left( \sqrt { 3 } , \frac { \pi } { 6 } , 1 \right)$ 
E)  $\left( 2 , \frac { \pi } { 3 } , \sqrt { 3 } \right)$ 
F) $\left( 2 , - \frac { \pi } { 3 } , \sqrt { 3 } \right)$ 
G) $\left( \sqrt { 3 } , \frac { \pi } { 3 } , 2 \right)$ 
H) $\left( \sqrt { 3 } , \frac { \pi } { 6 } , 2 \right)$ 
A)  $\left( 1 , \frac { \pi } { 3 } , \sqrt { 3 } \right)$ 
B)  $\left( 1 , \frac { \pi } { 6 } , \sqrt { 3 } \right)$ 
C)  $\left( \sqrt { 3 } , \frac { \pi } { 3 } , 1 \right)$ 
D)  $\left( \sqrt { 3 } , \frac { \pi } { 6 } , 1 \right)$ 
E)  $\left( 2 , \frac { \pi } { 3 } , \sqrt { 3 } \right)$ 
F) $\left( 2 , - \frac { \pi } { 3 } , \sqrt { 3 } \right)$ 
G) $\left( \sqrt { 3 } , \frac { \pi } { 3 } , 2 \right)$ 
H) $\left( \sqrt { 3 } , \frac { \pi } { 6 } , 2 \right)$

Question 16

Find the radius of the sphere whose equation is $x ^ { 2 } + 2 x + y ^ { 2 } + z ^ { 2 } = 4$ .

Accepted Answer

A) 1 
B)  $\sqrt { 2 }$ 
C)  $\sqrt { 3 }$ 
D) 2 
E)  $\sqrt { 5 }$ 
F) $\sqrt { 6 }$ 
G) $\sqrt { 7 }$ 
H) $2 \sqrt { 2 }$ 
A) 1 
B)  $\sqrt { 2 }$ 
C)  $\sqrt { 3 }$ 
D) 2 
E)  $\sqrt { 5 }$ 
F) $\sqrt { 6 }$ 
G) $\sqrt { 7 }$ 
H) $2 \sqrt { 2 }$

Question 17

Suppose u = $\langle 1,2,3 \rangle$ and v = $\langle 1,1 , - 2 \rangle$ . Find two vectors a and b such that u = a + b, a is parallel to v, and b is perpendicular to v.

Accepted Answer

The answer of Suppose u = $\langle 1,2,3 \rangle$ and...

Question 18

For the line l passing through the points (1, 0, 1) and (2, 4, 7), what is the value of c in the following symmetric equation for l? $\frac { x - 1 } { 1 } = \frac { y - 0 } { 4 } = \frac { z - 1 } { c }$

Accepted Answer

A) 1 
B) 2 
C) 3 
D) 4 
E) 5
F)6
G)7
H)8 
A) 1 
B) 2 
C) 3 
D) 4 
E) 5
F)6
G)7
H)8

Question 19

Given a = $\langle 1,0,1 \rangle$ and b = $\langle 0,1 , - 1 \rangle$ , find the vector projection of b onto a.

Accepted Answer

A)  $\langle 0,1 , - 1 \rangle$ 
B)  $\langle0 , - 1,1 \rangle$ 
C)  $\langle  0 , - \frac { 1 } { 2 } , \frac { 1 } { 2 } \rangle$ 
D)  $\langle 0 , \frac { 1 } { 2 } , - \frac { 1 } { 2 } \rangle$ 
E)  $(1,0,1)$ 
F) $\langle - 1,0 , - 1 \rangle$ 
G) $\langle - \frac { 1 } { 2 } , 0 , - \frac { 1 } { 2 } \rangle$ 
H) $\langle \frac { 1 } { 2 } , 0 , \frac { 1 } { 2 } \rangle$ 
A)  $\langle 0,1 , - 1 \rangle$ 
B)  $\langle0 , - 1,1 \rangle$ 
C)  $\langle  0 , - \frac { 1 } { 2 } , \frac { 1 } { 2 } \rangle$ 
D)  $\langle 0 , \frac { 1 } { 2 } , - \frac { 1 } { 2 } \rangle$ 
E)  $(1,0,1)$ 
F) $\langle - 1,0 , - 1 \rangle$ 
G) $\langle - \frac { 1 } { 2 } , 0 , - \frac { 1 } { 2 } \rangle$ 
H) $\langle \frac { 1 } { 2 } , 0 , \frac { 1 } { 2 } \rangle$

Question 20

Convert $( 1 , \pi , \pi )$ from spherical coordinates to rectangular coordinates.

Accepted Answer

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

Given $( - 1,2 )$ and $( 5,3 )$ , find their sum and illustrate geometrically.

Find the length of the vector 2i + j - 2k.

Determine whether the lines L₁: x = 1 + t, y = 2 + 3t, z = 3 + t and L₂: x = 1 + t, y = 3 + 4t, z = 4 + 2t are parallel, intersecting or skew. If they intersect, find the point of intersection.

Describe the surface whose equation in cylindrical coordinates is $\rho = 4 \cos \phi$ .

Find the range 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 x = 0.

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

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

Find a unit vector that has the same direction as the vector from the point P $( 1 , - 1,1 )$ to the point Q $( 3,2 , - 5 )$ .

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

Given $(0,3,2 )$ and $( 1,0 , - 3 )$ , find their sum and illustrate geometrically.

Let a = 3i - j + 2k and b = i - 2j + 3k. Find the scalar projection of a onto b.

Find an equation for the plane consisting of all points that are equidistant from the two points (1, 1, 0) and (0, 1, 1).

Find the cross product $\langle 1,2,0\rangle \times \langle 3,4,0 \rangle$ .

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

Find the radius of the sphere whose equation is $x ^ { 2 } + 2 x + y ^ { 2 } + z ^ { 2 } = 4$ .

Suppose u = $\langle 1,2,3 \rangle$ and v = $\langle 1,1 , - 2 \rangle$ . Find two vectors a and b such that u = a + b, a is parallel to v, and b is perpendicular to v.

For the line l passing through the points (1, 0, 1) and (2, 4, 7), what is the value of c in the following symmetric equation for l? $\frac { x - 1 } { 1 } = \frac { y - 0 } { 4 } = \frac { z - 1 } { c }$

Given a = $\langle 1,0,1 \rangle$ and b = $\langle 0,1 , - 1 \rangle$ , find the vector projection of b onto a.

Convert $( 1 , \pi , \pi )$ from spherical coordinates to rectangular coordinates.

Functions and Models

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Exam 9: Vectors and the Geometry of Space

Given (−1,2)( - 1,2 )(−1,2) and (5,3)( 5,3 )(5,3) , find their sum and illustrate geometrically.

Find the length of the vector 2i + j - 2k.

Determine whether the lines L1: x = 1 + t, y = 2 + 3t, z = 3 + t and L2: x = 1 + t, y = 3 + 4t, z = 4 + 2t are parallel, intersecting or skew. If they intersect, find the point of intersection.

Describe the surface whose equation in cylindrical coordinates is ρ=4cos⁡ϕ\rho = 4 \cos \phiρ=4cosϕ .

Find the range 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 x = 0.

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

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

Find a unit vector that has the same direction as the vector from the point P (1,−1,1)( 1 , - 1,1 )(1,−1,1) to the point Q (3,2,−5)( 3,2 , - 5 )(3,2,−5) .

Describe the surface whose equation in cylindrical coordinates is ρ=3sec⁡ϕ\rho = 3 \sec \phiρ=3secϕ .

Given (0,3,2)(0,3,2 )(0,3,2) and (1,0,−3)( 1,0 , - 3 )(1,0,−3) , find their sum and illustrate geometrically.

Let a = 3i - j + 2k and b = i - 2j + 3k. Find the scalar projection of a onto b.

Find an equation for the plane consisting of all points that are equidistant from the two points (1, 1, 0) and (0, 1, 1).

Find the cross product ⟨1,2,0⟩×⟨3,4,0⟩\langle 1,2,0\rangle \times \langle 3,4,0 \rangle⟨1,2,0⟩×⟨3,4,0⟩ .

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

Find the radius of the sphere whose equation is x2+2x+y2+z2=4x ^ { 2 } + 2 x + y ^ { 2 } + z ^ { 2 } = 4x2+2x+y2+z2=4 .

Suppose u = ⟨1,2,3⟩\langle 1,2,3 \rangle⟨1,2,3⟩ and v = ⟨1,1,−2⟩\langle 1,1 , - 2 \rangle⟨1,1,−2⟩ . Find two vectors a and b such that u = a + b, a is parallel to v, and b is perpendicular to v.

For the line l passing through the points (1, 0, 1) and (2, 4, 7), what is the value of c in the following symmetric equation for l? x−11=y−04=z−1c\frac { x - 1 } { 1 } = \frac { y - 0 } { 4 } = \frac { z - 1 } { c }1x−1​=4y−0​=cz−1​

Given a = ⟨1,0,1⟩\langle 1,0,1 \rangle⟨1,0,1⟩ and b = ⟨0,1,−1⟩\langle 0,1 , - 1 \rangle⟨0,1,−1⟩ , find the vector projection of b onto a.

Convert (1,π,π)( 1 , \pi , \pi )(1,π,π) from spherical coordinates to rectangular coordinates.

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

Given $( - 1,2 )$ and $( 5,3 )$ , find their sum and illustrate geometrically.

Determine whether the lines L₁: x = 1 + t, y = 2 + 3t, z = 3 + t and L₂: x = 1 + t, y = 3 + 4t, z = 4 + 2t are parallel, intersecting or skew. If they intersect, find the point of intersection.

Describe the surface whose equation in cylindrical coordinates is $\rho = 4 \cos \phi$ .

Find the range 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 x = 0.

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

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

Find a unit vector that has the same direction as the vector from the point P $( 1 , - 1,1 )$ to the point Q $( 3,2 , - 5 )$ .

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

Given $(0,3,2 )$ and $( 1,0 , - 3 )$ , find their sum and illustrate geometrically.

Find the cross product $\langle 1,2,0\rangle \times \langle 3,4,0 \rangle$ .

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

Find the radius of the sphere whose equation is $x ^ { 2 } + 2 x + y ^ { 2 } + z ^ { 2 } = 4$ .

Suppose u = $\langle 1,2,3 \rangle$ and v = $\langle 1,1 , - 2 \rangle$ . Find two vectors a and b such that u = a + b, a is parallel to v, and b is perpendicular to v.

For the line l passing through the points (1, 0, 1) and (2, 4, 7), what is the value of c in the following symmetric equation for l? $\frac { x - 1 } { 1 } = \frac { y - 0 } { 4 } = \frac { z - 1 } { c }$

Given a = $\langle 1,0,1 \rangle$ and b = $\langle 0,1 , - 1 \rangle$ , find the vector projection of b onto a.

Convert $( 1 , \pi , \pi )$ from spherical coordinates to rectangular coordinates.