Question 1

Describe the surface whose equation in cylindrical coordinates is z = 3.

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 2

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

Accepted Answer

A half cir

Question 3

Find the distance from the point (1, 2, 3) to the origin (0, 0, 0).

Accepted Answer

A)  $\sqrt { 13 }$ 
B)  $\sqrt { 14 }$ 
C)  $\sqrt { 15 }$ 
D) 4 
E)  $\sqrt { 17 }$ 
F) $\sqrt { 18 }$ 
G) $\sqrt { 19 }$ 
H) $\sqrt { 20 }$ 
A)  $\sqrt { 13 }$ 
B)  $\sqrt { 14 }$ 
C)  $\sqrt { 15 }$ 
D) 4 
E)  $\sqrt { 17 }$ 
F) $\sqrt { 18 }$ 
G) $\sqrt { 19 }$ 
H) $\sqrt { 20 }$

Question 4

Let S be the quadric surface given by $x ^ { 2 } + 2 y ^ { 2 } + 2 x - z = 0$ . What kind of surface is S?

Accepted Answer

The answer of Let S be the quadric surface given...

Question 5

Convert the point (0, -5, 0) to cylindrical and spherical coordinates.

Accepted Answer

Cylindrical coordina

Question 6

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

Accepted Answer

A half cir

Question 7

For the function $z = \sqrt { x ^ { 2 } + y ^ { 2 } - 1 }$ , sketch the portion of the surface in the first octant.

Accepted Answer

The answer of For the function \(z = \sqrt {...

Question 8

Find an equation of a plane containing the point P(2, -1, 1) and perpendicular to the z-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) $z = - 1$

Question 9

Find the distance from the point (4, -3, 6) to z-axis.

Accepted Answer

A)  $\sqrt { 61 }$ 
B)  $\sqrt { 52 }$ 
C) 6 
D) 0 
E) 5
F) $\sqrt { 45 }$ 
G)4
H)3 
A)  $\sqrt { 61 }$ 
B)  $\sqrt { 52 }$ 
C) 6 
D) 0 
E) 5
F) $\sqrt { 45 }$ 
G)4
H)3

Question 10

Let L be the line given by x = 2 - t, y = 1 + t, and z = 1 + 2t. L intersects the plane 2x + y - z = 1 at the point P = (1, 2, 3). Find the angle L makes with the plane, to the nearest degree.

Accepted Answer

The answer of Let L be the line given by...

Question 11

Find the cosine of the angle between the two vectors $\langle 1,1,1 \rangle$ and $\langle1,4,3 \rangle$ .

Accepted Answer

A)  $8 / \sqrt { 41 }$ 
B)  $8 / \sqrt { 53 }$ 
C)  $8 / \sqrt { 69 }$ 
D)  $8 / \sqrt { 78 }$ 
E)  $4 / \sqrt { 41 }$ 
F) $4 / \sqrt { 53 }$ 
G) $4 / \sqrt { 69 }$ 
H) $4 / \sqrt { 78 }$ 
A)  $8 / \sqrt { 41 }$ 
B)  $8 / \sqrt { 53 }$ 
C)  $8 / \sqrt { 69 }$ 
D)  $8 / \sqrt { 78 }$ 
E)  $4 / \sqrt { 41 }$ 
F) $4 / \sqrt { 53 }$ 
G) $4 / \sqrt { 69 }$ 
H) $4 / \sqrt { 78 }$

Question 12

Find the cosine of the angle between the two planes x + y + z = 0 and x + 2y + 3z = 1.

Accepted Answer

A) 4/ $\sqrt { 35 }$ 
B) 6/ $\sqrt { 35 }$ 
C) 7/ $\sqrt { 35 }$ 
D) 8/ $\sqrt { 35 }$ 
E) 5/ $\sqrt { 42 }$ 
F)6/ $\sqrt { 42 }$ 
G)7/ $\sqrt { 42 }$ 
H)8/ $\sqrt { 42 }$ 
A) 4/ $\sqrt { 35 }$ 
B) 6/ $\sqrt { 35 }$ 
C) 7/ $\sqrt { 35 }$ 
D) 8/ $\sqrt { 35 }$ 
E) 5/ $\sqrt { 42 }$ 
F)6/ $\sqrt { 42 }$ 
G)7/ $\sqrt { 42 }$ 
H)8/ $\sqrt { 42 }$

Question 13

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

Accepted Answer

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

Question 14

What value of x will cause the two vectors $\langle 5 x , - 1,2 \rangle$ and $\langle 3,2 x , - 2\rangle$ to be orthogonal?

Accepted Answer

A)  $- \frac { 17 } { 4 }$ 
B)  $- \frac { 4 } { 17 }$ 
C)  $\frac { 13 } { 4 }$ 
D)  $- \frac { 13 } { 4 }$ 
E)  $\frac { 17 } { 4 }$ 
F) $\frac { 4 } { 17 }$ 
G) $- \frac { 4 } { 13 }$ 
H) $\frac { 4 } { 13 }$ 
A)  $- \frac { 17 } { 4 }$ 
B)  $- \frac { 4 } { 17 }$ 
C)  $\frac { 13 } { 4 }$ 
D)  $- \frac { 13 } { 4 }$ 
E)  $\frac { 17 } { 4 }$ 
F) $\frac { 4 } { 17 }$ 
G) $- \frac { 4 } { 13 }$ 
H) $\frac { 4 } { 13 }$

Question 15

Find the cross product $\mathbf { a } \times \mathbf { b }$ ; where a = - i + 2j + 4k and b = 7i + 3j.

Accepted Answer

A) 15i - 8j + 10k 
B) 3i + 11j - 9k 
C) 7i + 13j + k 
D) 14i + 5j - 12k 
E) -7i - 23j + 33k
F)-10i + 27j + 13k
G)-12i + 28j - 17k
H)18i + 32j + 7k 
A) 15i - 8j + 10k 
B) 3i + 11j - 9k 
C) 7i + 13j + k 
D) 14i + 5j - 12k 
E) -7i - 23j + 33k
F)-10i + 27j + 13k
G)-12i + 28j - 17k
H)18i + 32j + 7k

Question 16

Find the lengths of the sides of the triangle ABC and determine whether the triangle is isosceles, a right triangle, both, or neither given: A(5, 5, 1), B(3, 3, 2), C(1, 4, 4).

Accepted Answer

AB = 3; BC

Question 17

If vectors a, b, and c are mutually orthogonal, find $\mathbf { a } \times ( \mathbf { b } \times \mathbf { c } )$ .

Accepted Answer

The answer of If vectors a, b, and c are...

Question 18

Describe the surface whose equation in cylindrical coordinates is $\theta = \frac { \pi } { 2 }$ .

Accepted Answer

The answer of Describe the surface whose equation in cylindrical...

Question 19

Let $\mathbf { u } \times \mathbf { v } = \mathbf { i } - \mathbf { j } + 2 \mathbf { k }$ . Find the area A of the parallelogram determined by u and v, if it exists.

Accepted Answer

The answer of Let \(\mathbf { u } \times \mathbf...

Question 20

Let a = 2i + j - k and b = i + 2j + 3k. Find an equation of the line parallel to a + b and passing through the tip of b.

Accepted Answer

r (t) = (1

Describe the surface whose equation in cylindrical coordinates is z = 3.

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

Find the distance from the point (1, 2, 3) to the origin (0, 0, 0).

Let S be the quadric surface given by $x ^ { 2 } + 2 y ^ { 2 } + 2 x - z = 0$ . What kind of surface is S?

Convert the point (0, -5, 0) to cylindrical and spherical coordinates.

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

For the function $z = \sqrt { x ^ { 2 } + y ^ { 2 } - 1 }$ , sketch the portion of the surface in the first octant. $For the function z = \sqrt { x ^ { 2 } + y ^ { 2 } - 1 } , sketch the portion of the surface in the first octant.$

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

Find the distance from the point (4, -3, 6) to z-axis.

Let L be the line given by x = 2 - t, y = 1 + t, and z = 1 + 2t. L intersects the plane 2x + y - z = 1 at the point P = (1, 2, 3). Find the angle L makes with the plane, to the nearest degree.

Find the cosine of the angle between the two vectors $\langle 1,1,1 \rangle$ and $\langle1,4,3 \rangle$ .

Find the cosine of the angle between the two planes x + y + z = 0 and x + 2y + 3z = 1.

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

What value of x will cause the two vectors $\langle 5 x , - 1,2 \rangle$ and $\langle 3,2 x , - 2\rangle$ to be orthogonal?

Find the cross product $\mathbf { a } \times \mathbf { b }$ ; where a = - i + 2j + 4k and b = 7i + 3j.

Find the lengths of the sides of the triangle ABC and determine whether the triangle is isosceles, a right triangle, both, or neither given: A(5, 5, 1), B(3, 3, 2), C(1, 4, 4).

If vectors a, b, and c are mutually orthogonal, find $\mathbf { a } \times ( \mathbf { b } \times \mathbf { c } )$ .

Describe the surface whose equation in cylindrical coordinates is $\theta = \frac { \pi } { 2 }$ .

Let $\mathbf { u } \times \mathbf { v } = \mathbf { i } - \mathbf { j } + 2 \mathbf { k }$ . Find the area A of the parallelogram determined by u and v, if it exists.

Let a = 2i + j - k and b = i + 2j + 3k. Find an equation of the line parallel to a + b and passing through the tip of b.

Functions and Models

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Exam 9: Vectors and the Geometry of Space

Describe the surface whose equation in cylindrical coordinates is z = 3.

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

Find the distance from the point (1, 2, 3) to the origin (0, 0, 0).

Let S be the quadric surface given by x2+2y2+2x−z=0x ^ { 2 } + 2 y ^ { 2 } + 2 x - z = 0x2+2y2+2x−z=0 . What kind of surface is S?

Convert the point (0, -5, 0) to cylindrical and spherical coordinates.

Find the set of intersection of the surfaces whose equations in spherical coordinates are ρ=4\rho = 4ρ=4 and θ=π4\theta = \frac { \pi } { 4 }θ=4π​ .

For the function z=x2+y2−1z = \sqrt { x ^ { 2 } + y ^ { 2 } - 1 }z=x2+y2−1​ , sketch the portion of the surface in the first octant.

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

Find the distance from the point (4, -3, 6) to z-axis.

Let L be the line given by x = 2 - t, y = 1 + t, and z = 1 + 2t. L intersects the plane 2x + y - z = 1 at the point P = (1, 2, 3). Find the angle L makes with the plane, to the nearest degree.

Find the cosine of the angle between the two vectors ⟨1,1,1⟩\langle 1,1,1 \rangle⟨1,1,1⟩ and ⟨1,4,3⟩\langle1,4,3 \rangle⟨1,4,3⟩ .

Find the cosine of the angle between the two planes x + y + z = 0 and x + 2y + 3z = 1.

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

What value of x will cause the two vectors ⟨5x,−1,2⟩\langle 5 x , - 1,2 \rangle⟨5x,−1,2⟩ and ⟨3,2x,−2⟩\langle 3,2 x , - 2\rangle⟨3,2x,−2⟩ to be orthogonal?

Find the cross product a×b\mathbf { a } \times \mathbf { b }a×b ; where a = - i + 2j + 4k and b = 7i + 3j.

Find the lengths of the sides of the triangle ABC and determine whether the triangle is isosceles, a right triangle, both, or neither given: A(5, 5, 1), B(3, 3, 2), C(1, 4, 4).

If vectors a, b, and c are mutually orthogonal, find a×(b×c)\mathbf { a } \times ( \mathbf { b } \times \mathbf { c } )a×(b×c) .

Describe the surface whose equation in cylindrical coordinates is θ=π2\theta = \frac { \pi } { 2 }θ=2π​ .

Let u×v=i−j+2k\mathbf { u } \times \mathbf { v } = \mathbf { i } - \mathbf { j } + 2 \mathbf { k }u×v=i−j+2k . Find the area A of the parallelogram determined by u and v, if it exists.

Let a = 2i + j - k and b = i + 2j + 3k. Find an equation of the line parallel to a + b and passing through the tip of b.

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 $\theta = 0$ .

Let S be the quadric surface given by $x ^ { 2 } + 2 y ^ { 2 } + 2 x - z = 0$ . What kind of surface is S?

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

For the function $z = \sqrt { x ^ { 2 } + y ^ { 2 } - 1 }$ , sketch the portion of the surface in the first octant. $For the function z = \sqrt { x ^ { 2 } + y ^ { 2 } - 1 } , sketch the portion of the surface in the first octant.$

Find the cosine of the angle between the two vectors $\langle 1,1,1 \rangle$ and $\langle1,4,3 \rangle$ .

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

What value of x will cause the two vectors $\langle 5 x , - 1,2 \rangle$ and $\langle 3,2 x , - 2\rangle$ to be orthogonal?

Find the cross product $\mathbf { a } \times \mathbf { b }$ ; where a = - i + 2j + 4k and b = 7i + 3j.

If vectors a, b, and c are mutually orthogonal, find $\mathbf { a } \times ( \mathbf { b } \times \mathbf { c } )$ .

Describe the surface whose equation in cylindrical coordinates is $\theta = \frac { \pi } { 2 }$ .

Let $\mathbf { u } \times \mathbf { v } = \mathbf { i } - \mathbf { j } + 2 \mathbf { k }$ . Find the area A of the parallelogram determined by u and v, if it exists.