Question 1

Let f(x, y) = x sin y. If x = $\pi$ , find f(x, x/2).

Accepted Answer

A)  $\frac { \pi } { 6 }$ 
B)  $\frac { \pi } { 4 }$ 
C)  $\frac { \pi } { 3 }$ 
D)  $\frac { \pi } { 2 }$ 
E)  $\frac { 2 \pi } { 3 }$ 
F) $\frac { 3 \pi } { 4 }$ 
G) $\pi$ 
H) $2 \pi$ 
A)  $\frac { \pi } { 6 }$ 
B)  $\frac { \pi } { 4 }$ 
C)  $\frac { \pi } { 3 }$ 
D)  $\frac { \pi } { 2 }$ 
E)  $\frac { 2 \pi } { 3 }$ 
F) $\frac { 3 \pi } { 4 }$ 
G) $\pi$ 
H) $2 \pi$

Question 2

Find the coordinate of the initial point of a vector $\mathbf { a } = \langle 4 , - 3 \rangle$ whose terminal point is $( 5 , - 2 )$ .

Accepted Answer

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

Question 3

Find $\mathbf { a } \times \mathbf { b }$ , where a and b are given in the figure.

Accepted Answer

The answer of Find \(\mathbf { a } \times \mathbf...

Question 4

Find an equation of a plane containing the point P(2, -1, 1) and perpendicular to the x-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 5

Find the equation of the plane containing the lines x = 4 - 4t, y = 3 - t, z = 1 + 5t and x = 4 - t, y = 3 + 2t, z = 1.

Accepted Answer

10x - 5y +

Question 6

Find the area of the triangle whose vertices are A(1,-1, 2), B(4, 0, 1), and C(2,-1, 2).

Accepted Answer

A)  $\frac { \sqrt { 30 } } { 2 }$ 
B)  $\frac { \sqrt { 3 } } { 2 }$ 
C)  $\frac { \sqrt { 2 } } { 2 }$ 
D)  $| \sqrt { 2 } |$ 
E)  $\frac { 4 } { 3 }$ 
F) $\sqrt { 30 }$ 
G)3
H) $\sqrt { 3 }$ 
A)  $\frac { \sqrt { 30 } } { 2 }$ 
B)  $\frac { \sqrt { 3 } } { 2 }$ 
C)  $\frac { \sqrt { 2 } } { 2 }$ 
D)  $| \sqrt { 2 } |$ 
E)  $\frac { 4 } { 3 }$ 
F) $\sqrt { 30 }$ 
G)3
H) $\sqrt { 3 }$

Question 7

Let S be the quadric surface given by $x ^ { 2 } + 2 y ^ { 2 } + 2 x - z = 0$ . What are the traces of S in each of the three coordinate planes?

Accepted Answer

xy-plane: ellipse; y

Question 8

Find the distance between the two skew lines $L _ { 1 } : \frac { x - 1 } { 2 } = \frac { y - 3 } { 5 } = \frac { z - 1 } { 3 }$ and $L _ { 2 } : \frac { x - 2 } { 4 } = \frac { y + 1 } { 2 } = \frac { z + 2 } { 3 }$

Accepted Answer

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

Question 9

Give a counterexample showing that the statement "If $\mathbf { a } \times \mathbf { b } = \mathbf { a } \times \mathbf { c }$ , then b = c" is not always true.

Accepted Answer

For example, if a =

Question 10

Find the distance from the point (4, -3, 5) to xz-plane.

Accepted Answer

A) -5 
B)  $\sqrt { 34 }$ 
C) -4 
D) -3 
E) 5
F) $\sqrt { 50 }$ 
G)4
H)3 
A) -5 
B)  $\sqrt { 34 }$ 
C) -4 
D) -3 
E) 5
F) $\sqrt { 50 }$ 
G)4
H)3

Question 11

Identify the trace of the surface $x = y ^ { 2 } + z ^ { 2 }$ 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 12

Find the angle between the lines $\frac { x - 2 } { 1 } = \frac { 1 - y } { 3 } = \frac { z - 3 } { 1 }$ and $\frac { x } { 2 } = \frac { y + 3 } { - 1 } = \frac { z - 1 } { 2 }$ .

Accepted Answer

The answer of Find the angle between the lines \(\frac...

Question 13

Find the length of the cross product of the vectors $\langle 1,1,1 \rangle$ and $\langle1,1,2 \rangle$ .

Accepted Answer

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

Question 14

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

Accepted Answer

A)  $y + z = 0$ 
B)  $x - y = 3$ 
C)  $x = 2$ 
D)  $x - 2 z = 0$ 
E)  $y - z = 0$ 
F) $x + 2 y = 0$ 
G) $z = 1$ 
H) $z = - 1$ 
A)  $y + z = 0$ 
B)  $x - y = 3$ 
C)  $x = 2$ 
D)  $x - 2 z = 0$ 
E)  $y - z = 0$ 
F) $x + 2 y = 0$ 
G) $z = 1$ 
H) $z = - 1$

Question 15

Is the quadrilateral ABCD formed by the points A $( - 1,2,0 )$ , B $( 3,1,1 )$ , C $( 1,2,1 )$ , and D $( - 3,3,0 )$ a parallelogram?

Accepted Answer

The answer of Is the quadrilateral ABCD formed by the...

Question 16

Find $| \mathbf { a } \times \mathbf { b } |$ , given that $| \mathbf { a } | = 2$ , $| \mathbf { b } | = 5$ , and the angle between a and b is $\frac { 5 \pi } { 6 }$ .

Accepted Answer

A)  $5 \sqrt { 5 }$ 
B)  $5 \sqrt { 2 }$ 
C) 10 
D) 5 
E)  $5 \sqrt { 3 }$ 
F) $\frac { 5 \sqrt { 2 } } { 2 }$ 
G) $- 5 \sqrt { 5 }$ 
H)-5 
A)  $5 \sqrt { 5 }$ 
B)  $5 \sqrt { 2 }$ 
C) 10 
D) 5 
E)  $5 \sqrt { 3 }$ 
F) $\frac { 5 \sqrt { 2 } } { 2 }$ 
G) $- 5 \sqrt { 5 }$ 
H)-5

Question 17

Find an equation of the plane passing through the points A, B, and C shown below.

Accepted Answer

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

Question 18

Describe the vertical traces x = k and y = k and the horizontal traces z = k for the function (x, y) = $x ^ { 2 } - y ^ { 2 }$ .

Accepted Answer

x = k: @#IMG-DLM& , parabola;

Question 19

Let a = $\langle - 1,2,2 \rangle$ and suppose that b is a vector parallel to a. Find projb a.

Accepted Answer

Question 20

Consider the solid sphere $( x - 1 ) ^ { 2 } + ( y + 1 ) ^ { 2 } + ( z - 2 ) ^ { 2 } \leq 9$ . Which of the following points are inside the sphere?
(i) (0, 0, 0)
(ii) (2, 2, 2)
(iii) $( 3 , - 1,1 )$ (iv) $( - 1,1,1 )$

Accepted Answer

(i) inside (ii) outs

Let f(x, y) = x sin y. If x = $\pi$ , find f(x, x/2).

Find the coordinate of the initial point of a vector $\mathbf { a } = \langle 4 , - 3 \rangle$ whose terminal point is $( 5 , - 2 )$ .

Find $\mathbf { a } \times \mathbf { b }$ , where a and b are given in the figure. $Find \mathbf { a } \times \mathbf { b } , where a and b are given in the figure.$

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

Find the equation of the plane containing the lines x = 4 - 4t, y = 3 - t, z = 1 + 5t and x = 4 - t, y = 3 + 2t, z = 1.

Find the area of the triangle whose vertices are A(1,-1, 2), B(4, 0, 1), and C(2,-1, 2).

Let S be the quadric surface given by $x ^ { 2 } + 2 y ^ { 2 } + 2 x - z = 0$ . What are the traces of S in each of the three coordinate planes?

Find the distance between the two skew lines $L _ { 1 } : \frac { x - 1 } { 2 } = \frac { y - 3 } { 5 } = \frac { z - 1 } { 3 }$ and $L _ { 2 } : \frac { x - 2 } { 4 } = \frac { y + 1 } { 2 } = \frac { z + 2 } { 3 }$

Give a counterexample showing that the statement "If $\mathbf { a } \times \mathbf { b } = \mathbf { a } \times \mathbf { c }$ , then b = c" is not always true.

Find the distance from the point (4, -3, 5) to xz-plane.

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

Find the angle between the lines $\frac { x - 2 } { 1 } = \frac { 1 - y } { 3 } = \frac { z - 3 } { 1 }$ and $\frac { x } { 2 } = \frac { y + 3 } { - 1 } = \frac { z - 1 } { 2 }$ .

Find the length of the cross product of the vectors $\langle 1,1,1 \rangle$ and $\langle1,1,2 \rangle$ .

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

Is the quadrilateral ABCD formed by the points A $( - 1,2,0 )$ , B $( 3,1,1 )$ , C $( 1,2,1 )$ , and D $( - 3,3,0 )$ a parallelogram?

Find $| \mathbf { a } \times \mathbf { b } |$ , given that $| \mathbf { a } | = 2$ , $| \mathbf { b } | = 5$ , and the angle between a and b is $\frac { 5 \pi } { 6 }$ .

Find an equation of the plane passing through the points A, B, and C shown below.

Describe the vertical traces x = k and y = k and the horizontal traces z = k for the function (x, y) = $x ^ { 2 } - y ^ { 2 }$ .

Let a = $\langle - 1,2,2 \rangle$ and suppose that b is a vector parallel to a. Find proj_b a.

Consider the solid sphere $( x - 1 ) ^ { 2 } + ( y + 1 ) ^ { 2 } + ( z - 2 ) ^ { 2 } \leq 9$ . Which of the following points are inside the sphere? (i) (0, 0, 0) (ii) (2, 2, 2) (iii) $( 3 , - 1,1 )$ (iv) $( - 1,1,1 )$

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

Let f(x, y) = x sin y. If x = π\piπ , find f(x, x/2).

Find the coordinate of the initial point of a vector a=⟨4,−3⟩\mathbf { a } = \langle 4 , - 3 \ranglea=⟨4,−3⟩ whose terminal point is (5,−2)( 5 , - 2 )(5,−2) .

Find a×b\mathbf { a } \times \mathbf { b }a×b , where a and b are given in the figure.

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

Find the equation of the plane containing the lines x = 4 - 4t, y = 3 - t, z = 1 + 5t and x = 4 - t, y = 3 + 2t, z = 1.

Find the area of the triangle whose vertices are A(1,-1, 2), B(4, 0, 1), and C(2,-1, 2).

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 are the traces of S in each of the three coordinate planes?

Give a counterexample showing that the statement "If a×b=a×c\mathbf { a } \times \mathbf { b } = \mathbf { a } \times \mathbf { c }a×b=a×c , then b = c" is not always true.

Find the distance from the point (4, -3, 5) to xz-plane.

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

Find the angle between the lines x−21=1−y3=z−31\frac { x - 2 } { 1 } = \frac { 1 - y } { 3 } = \frac { z - 3 } { 1 }1x−2​=31−y​=1z−3​ and x2=y+3−1=z−12\frac { x } { 2 } = \frac { y + 3 } { - 1 } = \frac { z - 1 } { 2 }2x​=−1y+3​=2z−1​ .

Find the length of the cross product of the vectors ⟨1,1,1⟩\langle 1,1,1 \rangle⟨1,1,1⟩ and ⟨1,1,2⟩\langle1,1,2 \rangle⟨1,1,2⟩ .

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

Is the quadrilateral ABCD formed by the points A (−1,2,0)( - 1,2,0 )(−1,2,0) , B (3,1,1)( 3,1,1 )(3,1,1) , C (1,2,1)( 1,2,1 )(1,2,1) , and D (−3,3,0)( - 3,3,0 )(−3,3,0) a parallelogram?

Find ∣a×b∣| \mathbf { a } \times \mathbf { b } |∣a×b∣ , given that ∣a∣=2| \mathbf { a } | = 2∣a∣=2 , ∣b∣=5| \mathbf { b } | = 5∣b∣=5 , and the angle between a and b is 5π6\frac { 5 \pi } { 6 }65π​ .

Find an equation of the plane passing through the points A, B, and C shown below.

Describe the vertical traces x = k and y = k and the horizontal traces z = k for the function (x, y) = x2−y2x ^ { 2 } - y ^ { 2 }x2−y2 .

Let a = ⟨−1,2,2⟩\langle - 1,2,2 \rangle⟨−1,2,2⟩ and suppose that b is a vector parallel to a. Find projb a.

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

Let f(x, y) = x sin y. If x = $\pi$ , find f(x, x/2).

Find the coordinate of the initial point of a vector $\mathbf { a } = \langle 4 , - 3 \rangle$ whose terminal point is $( 5 , - 2 )$ .

Find $\mathbf { a } \times \mathbf { b }$ , where a and b are given in the figure. $Find \mathbf { a } \times \mathbf { b } , where a and b are given in the figure.$

Let S be the quadric surface given by $x ^ { 2 } + 2 y ^ { 2 } + 2 x - z = 0$ . What are the traces of S in each of the three coordinate planes?

Give a counterexample showing that the statement "If $\mathbf { a } \times \mathbf { b } = \mathbf { a } \times \mathbf { c }$ , then b = c" is not always true.

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

Find the angle between the lines $\frac { x - 2 } { 1 } = \frac { 1 - y } { 3 } = \frac { z - 3 } { 1 }$ and $\frac { x } { 2 } = \frac { y + 3 } { - 1 } = \frac { z - 1 } { 2 }$ .

Find the length of the cross product of the vectors $\langle 1,1,1 \rangle$ and $\langle1,1,2 \rangle$ .

Is the quadrilateral ABCD formed by the points A $( - 1,2,0 )$ , B $( 3,1,1 )$ , C $( 1,2,1 )$ , and D $( - 3,3,0 )$ a parallelogram?

Find $| \mathbf { a } \times \mathbf { b } |$ , given that $| \mathbf { a } | = 2$ , $| \mathbf { b } | = 5$ , and the angle between a and b is $\frac { 5 \pi } { 6 }$ .

Describe the vertical traces x = k and y = k and the horizontal traces z = k for the function (x, y) = $x ^ { 2 } - y ^ { 2 }$ .

Let a = $\langle - 1,2,2 \rangle$ and suppose that b is a vector parallel to a. Find proj_b a.