Question 1

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

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 2

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

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 3

Given $\mathbf { a } = \langle 1,1 \rangle$ , $\mathbf { b } = \langle - 4,2 \rangle$ , and $\mathbf { c } = \langle 5,2 \rangle$ , find s and t such that c = sa + tb.

Accepted Answer

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

Question 4

Identify the surface $x ^ { 2 } - y ^ { 2 } + z ^ { 2 } = 10$ .

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 5

Find the cosine of the angle between the two vectors $\langle 1,2 \rangle$ and $\langle - 2,1 \rangle$ .

Accepted Answer

A)  $\frac { 1 } { \sqrt { 2 } }$ 
B) 0 
C)  $\frac { 1 } { \sqrt { 3 } }$ 
D) 1 
E)  $\frac { 1 } { \sqrt { 5 } }$ 
F) $\frac { 1 } { 2 }$ 
G) $\frac { 4 } { 3 }$ 
H) $\frac { 3 } { 4 }$ 
A)  $\frac { 1 } { \sqrt { 2 } }$ 
B) 0 
C)  $\frac { 1 } { \sqrt { 3 } }$ 
D) 1 
E)  $\frac { 1 } { \sqrt { 5 } }$ 
F) $\frac { 1 } { 2 }$ 
G) $\frac { 4 } { 3 }$ 
H) $\frac { 3 } { 4 }$

Question 6

If we know that $\mathbf { a } \times \mathbf { b } = \mathbf { a } \times \mathbf { c }$ , which of the following statements is true?

Accepted Answer

A) b and c are parallel 
B)  $| \mathbf { b } | = | \mathbf { c } |$ 
C) b = c 
D) a is parallel to b - c 
E) a is a unit vector 
A) b and c are parallel 
B)  $| \mathbf { b } | = | \mathbf { c } |$ 
C) b = c 
D) a is parallel to b - c 
E) a is a unit vector

Question 7

Find the distance from the point P(-1, 0, 2) to the plane passing through the points A(-2, 1, 1), B(0, 5, 2), and C(1, 3, 0).

Accepted Answer

The answer of Find the distance from the point P(-1,...

Question 8

A suitcase is pulled along a straight path on the floor by a rope that applies a force of 70 lb at an angle of 60° with the floor. How much work is done in moving the suitcase 20 ft?

Accepted Answer

The answer of A suitcase is pulled along a straight...

Question 9

Identify the trace of the surface $x ^ { 2 } = y ^ { 2 } + z ^ { 2 }$ in the plane z = 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 10

Let f(x, y) = $\sqrt { x } + \sqrt { y }$ (a) Evaluate f (4, 9).(b) Find the domain of f.(c) Find the range of f.

Accepted Answer

(a) 5
(b)

Question 11

Find an equation of the sphere with center P $( 6 , - 2,4 )$ that is tangent to the xz-plane.

Accepted Answer

The answer of Find an equation of the sphere with...

Question 12

Let v and w be vectors which are both perpendicular to $\langle 1,3 , - 1 \rangle$ . Assuming that $\langle 0 , - 2,1 \rangle$ = av + bw + c $\langle 1,3 , - 1 \rangle$ , find c.

Accepted Answer

The answer of Let v and w be vectors which...

Question 13

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

Accepted Answer

The answer of Determine whether the points A(2, 2, 4),...

Question 14

Given three points P(1, -1, 0), Q(0, 1, 2) and R(-1, -1, 1), find the distance from the point Q to the line passing through the points P and R.

Accepted Answer

A)  $\sqrt { 29 }$ 
B)  $\sqrt { \frac { 29 } { 5 } }$ 
C)  $\sqrt { \frac { 29 } { 6 } }$ 
D)  $\frac { \sqrt { 29 } } { 3 }$ 
E)  $\frac { 1 } { \sqrt { 29 } }$ 
F) $\sqrt { \frac { 5 } { 29 } }$ 
G) $\sqrt { \frac { 6 } { 29 } }$ 
H) $\frac { 3 } { \sqrt { 29 } }$ 
A)  $\sqrt { 29 }$ 
B)  $\sqrt { \frac { 29 } { 5 } }$ 
C)  $\sqrt { \frac { 29 } { 6 } }$ 
D)  $\frac { \sqrt { 29 } } { 3 }$ 
E)  $\frac { 1 } { \sqrt { 29 } }$ 
F) $\sqrt { \frac { 5 } { 29 } }$ 
G) $\sqrt { \frac { 6 } { 29 } }$ 
H) $\frac { 3 } { \sqrt { 29 } }$

Question 15

Use the given data:
Los Angeles: Latitude 34.05°N and Longitude 118.25°W;
Hawaii: Latitude 21.3°N and Longitude 157.83°W.Find the distance from Los Angeles to Hawaii (Assume the radius of earth is 3960 miles.)

Accepted Answer

The answer of Use the given data:
Los Angeles: Latitude 34.05°N...

Question 16

Find the components of a vector a whose initial point is $( 1 , - 1,3 )$ and terminal point is $( 4 , - 3,5 )$ .

Accepted Answer

A)  $\{ 1 , - 1,3 \}$ 
B)  $\{ 4 , - 3,5 \}$ 
C)  $\{ 3 , - 2,2 \}$ 
D)  $\{ 5 , - 4,8 \}$ 
E)  $\{ - 1,1 , - 3 \}$ 
F) $\{ - 4,3 , - 5 \}$ 
G) $( - 3,2 , - 2 \}$ 
H) $( - 5,4 , - 8 \}$ 
A)  $\{ 1 , - 1,3 \}$ 
B)  $\{ 4 , - 3,5 \}$ 
C)  $\{ 3 , - 2,2 \}$ 
D)  $\{ 5 , - 4,8 \}$ 
E)  $\{ - 1,1 , - 3 \}$ 
F) $\{ - 4,3 , - 5 \}$ 
G) $( - 3,2 , - 2 \}$ 
H) $( - 5,4 , - 8 \}$

Question 17

Sketch the solid given in cylindrical coordinates by $0 \leq \theta \leq \frac { \pi } { 2 } , 1 \leq r \leq 3 , r \leq z \leq 3$ .

Accepted Answer

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

Question 18

Find the coordinates of the point halfway between the midpoints of the vectors a = 3i - 5j + k and b = 5i + 3j + k.

Accepted Answer

The answer of Find the coordinates of the point halfway...

Question 19

Find the volume of the parallelepiped with adjacent edges PQ, PR, and PS, where P (-2, 1, 1), Q(-1, 0, 2), R(0, 5, 2), and S(1, 3, 0).

Accepted Answer

A) 19 
B) 22 
C) 38 
D) 80 
E) 20
F)40
G)44
H)72 
A) 19 
B) 22 
C) 38 
D) 80 
E) 20
F)40
G)44
H)72

Question 20

Find the distance from the origin to the plane x + 2y + 2z = -6.

Accepted Answer

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

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

H)5

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

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

Given $\mathbf { a } = \langle 1,1 \rangle$ , $\mathbf { b } = \langle - 4,2 \rangle$ , and $\mathbf { c } = \langle 5,2 \rangle$ , find s and t such that c = sa + tb.

Identify the surface $x ^ { 2 } - y ^ { 2 } + z ^ { 2 } = 10$ .

Find the cosine of the angle between the two vectors $\langle 1,2 \rangle$ and $\langle - 2,1 \rangle$ .

If we know that $\mathbf { a } \times \mathbf { b } = \mathbf { a } \times \mathbf { c }$ , which of the following statements is true?

Find the distance from the point P(-1, 0, 2) to the plane passing through the points A(-2, 1, 1), B(0, 5, 2), and C(1, 3, 0).

A suitcase is pulled along a straight path on the floor by a rope that applies a force of 70 lb at an angle of 60° with the floor. How much work is done in moving the suitcase 20 ft?

Identify the trace of the surface $x ^ { 2 } = y ^ { 2 } + z ^ { 2 }$ in the plane z = 0.

Let f(x, y) = $\sqrt { x } + \sqrt { y }$ (a) Evaluate f (4, 9).(b) Find the domain of f.(c) Find the range of f.

Find an equation of the sphere with center P $( 6 , - 2,4 )$ that is tangent to the xz-plane.

Let v and w be vectors which are both perpendicular to $\langle 1,3 , - 1 \rangle$ . Assuming that $\langle 0 , - 2,1 \rangle$ = av + bw + c $\langle 1,3 , - 1 \rangle$ , find c.

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

Given three points P(1, -1, 0), Q(0, 1, 2) and R(-1, -1, 1), find the distance from the point Q to the line passing through the points P and R.

Use the given data: Los Angeles: Latitude 34.05°N and Longitude 118.25°W; Hawaii: Latitude 21.3°N and Longitude 157.83°W.Find the distance from Los Angeles to Hawaii (Assume the radius of earth is 3960 miles.)

Find the components of a vector a whose initial point is $( 1 , - 1,3 )$ and terminal point is $( 4 , - 3,5 )$ .

Sketch the solid given in cylindrical coordinates by $0 \leq \theta \leq \frac { \pi } { 2 } , 1 \leq r \leq 3 , r \leq z \leq 3$ . $Sketch the solid given in cylindrical coordinates by 0 \leq \theta \leq \frac { \pi } { 2 } , 1 \leq r \leq 3 , r \leq z \leq 3 .$

Find the coordinates of the point halfway between the midpoints of the vectors a = 3i - 5j + k and b = 5i + 3j + k.

Find the volume of the parallelepiped with adjacent edges PQ, PR, and PS, where P (-2, 1, 1), Q(-1, 0, 2), R(0, 5, 2), and S(1, 3, 0).

Find the distance from the origin to the plane x + 2y + 2z = -6.

Functions and Models

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Exam 9: Vectors and the Geometry of Space

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

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

Given a=⟨1,1⟩\mathbf { a } = \langle 1,1 \ranglea=⟨1,1⟩ , b=⟨−4,2⟩\mathbf { b } = \langle - 4,2 \rangleb=⟨−4,2⟩ , and c=⟨5,2⟩\mathbf { c } = \langle 5,2 \ranglec=⟨5,2⟩ , find s and t such that c = sa + tb.

Identify the surface x2−y2+z2=10x ^ { 2 } - y ^ { 2 } + z ^ { 2 } = 10x2−y2+z2=10 .

Find the cosine of the angle between the two vectors ⟨1,2⟩\langle 1,2 \rangle⟨1,2⟩ and ⟨−2,1⟩\langle - 2,1 \rangle⟨−2,1⟩ .

If we know that a×b=a×c\mathbf { a } \times \mathbf { b } = \mathbf { a } \times \mathbf { c }a×b=a×c , which of the following statements is true?

Find the distance from the point P(-1, 0, 2) to the plane passing through the points A(-2, 1, 1), B(0, 5, 2), and C(1, 3, 0).

A suitcase is pulled along a straight path on the floor by a rope that applies a force of 70 lb at an angle of 60° with the floor. How much work is done in moving the suitcase 20 ft?

Identify the trace of the surface x2=y2+z2x ^ { 2 } = y ^ { 2 } + z ^ { 2 }x2=y2+z2 in the plane z = 0.

Let f(x, y) = x+y\sqrt { x } + \sqrt { y }x​+y​ (a) Evaluate f (4, 9).(b) Find the domain of f.(c) Find the range of f.

Find an equation of the sphere with center P (6,−2,4)( 6 , - 2,4 )(6,−2,4) that is tangent to the xz-plane.

Let v and w be vectors which are both perpendicular to ⟨1,3,−1⟩\langle 1,3 , - 1 \rangle⟨1,3,−1⟩ . Assuming that ⟨0,−2,1⟩\langle 0 , - 2,1 \rangle⟨0,−2,1⟩ = av + bw + c ⟨1,3,−1⟩\langle 1,3 , - 1 \rangle⟨1,3,−1⟩ , find c.

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

Given three points P(1, -1, 0), Q(0, 1, 2) and R(-1, -1, 1), find the distance from the point Q to the line passing through the points P and R.

Use the given data: Los Angeles: Latitude 34.05°N and Longitude 118.25°W; Hawaii: Latitude 21.3°N and Longitude 157.83°W.Find the distance from Los Angeles to Hawaii (Assume the radius of earth is 3960 miles.)

Find the components of a vector a whose initial point is (1,−1,3)( 1 , - 1,3 )(1,−1,3) and terminal point is (4,−3,5)( 4 , - 3,5 )(4,−3,5) .

Sketch the solid given in cylindrical coordinates by 0≤θ≤π2,1≤r≤3,r≤z≤30 \leq \theta \leq \frac { \pi } { 2 } , 1 \leq r \leq 3 , r \leq z \leq 30≤θ≤2π​,1≤r≤3,r≤z≤3 .

Find the coordinates of the point halfway between the midpoints of the vectors a = 3i - 5j + k and b = 5i + 3j + k.

Find the volume of the parallelepiped with adjacent edges PQ, PR, and PS, where P (-2, 1, 1), Q(-1, 0, 2), R(0, 5, 2), and S(1, 3, 0).

Find the distance from the origin to the plane x + 2y + 2z = -6.

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

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

Given $\mathbf { a } = \langle 1,1 \rangle$ , $\mathbf { b } = \langle - 4,2 \rangle$ , and $\mathbf { c } = \langle 5,2 \rangle$ , find s and t such that c = sa + tb.

Identify the surface $x ^ { 2 } - y ^ { 2 } + z ^ { 2 } = 10$ .

Find the cosine of the angle between the two vectors $\langle 1,2 \rangle$ and $\langle - 2,1 \rangle$ .

If we know that $\mathbf { a } \times \mathbf { b } = \mathbf { a } \times \mathbf { c }$ , which of the following statements is true?

Identify the trace of the surface $x ^ { 2 } = y ^ { 2 } + z ^ { 2 }$ in the plane z = 0.

Let f(x, y) = $\sqrt { x } + \sqrt { y }$ (a) Evaluate f (4, 9).(b) Find the domain of f.(c) Find the range of f.

Find an equation of the sphere with center P $( 6 , - 2,4 )$ that is tangent to the xz-plane.

Let v and w be vectors which are both perpendicular to $\langle 1,3 , - 1 \rangle$ . Assuming that $\langle 0 , - 2,1 \rangle$ = av + bw + c $\langle 1,3 , - 1 \rangle$ , find c.

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

Find the components of a vector a whose initial point is $( 1 , - 1,3 )$ and terminal point is $( 4 , - 3,5 )$ .

Sketch the solid given in cylindrical coordinates by $0 \leq \theta \leq \frac { \pi } { 2 } , 1 \leq r \leq 3 , r \leq z \leq 3$ . $Sketch the solid given in cylindrical coordinates by 0 \leq \theta \leq \frac { \pi } { 2 } , 1 \leq r \leq 3 , r \leq z \leq 3 .$