Name Class. Date Section. Test Form A Chapter 11. Chapter 11 Test Bank 155

(1)

Name

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Date

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Class

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Section

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Test Form A Chapter 11

1. Find a unit vector in the direction of v if v is the vector from

(a) (b) (c)

(d) (e) None of these

2. Calculate

(a) (b) (c) (d) (e) None of these

In problems 3 through 8, let

3. Calculate

(a) (b) (c)

(d) 1 (e) None of these

4. Calculate

(a) (b) (c)

5. Calculate

(a) (b) (c)

6. Calculate

(a) (b) (c)

7. Calculate is the angle between

(a) (b) (c) (d) (e) None of these

8. Which of the following vectors is orthogonal to both

(a) (b) (c)

(d) w (e) None of these

v u

8i 2j 11k k

u and v?

13 217

13

238

7

238

7 217

u and v.

cos where

14, 21, 14

17

¹⁴¹⁷^,²¹¹⁷^,¹⁴¹⁷

17⁷

proj_vu.

24

16

18i 6j 6k

30

u^v^w^.

4, 10, 11

7

8, 2, 11

4, 10, 11

u v.

8, 2, 11

13

7 u^v.

u 3i j 2k, v 2i 3j 2k and w i 2k.

i

k k

j j i.

1

3i 1

3j 1

3k

1

3i 1

3j 1

3k i j k

3i 3j 3k

P1, 2, 3 to Q4, 1, 0.

(2)

9. Which of the following is an orthogonal pair of vectors?

(a) (b) (c)

10. Which of the following is a set of parametric equations for the line through the points and

(a) (b) (c)

11. Write an equation of the plane that contains the line given by and is perpendicular to the line given by

(a) (b) (c)

12. Which of the following is a sketch of the plane given by

(a) (b)

(c) (d)

(e) None of these

13. Calculate the distance from the point to the line given by

(a) (b) (c)

(d)

³² (e) None of these

5

14

¹⁴³

5

2

and z 1 t. 1, 2, 3 x 2 t, y 1,

x y

z

2

6 4

4 6

6

x y

z

2 4 4

4 6

6

x y

z

2 4

4 6

6

x y

z

1 2 1

3 1

2 3

3

2x y 3z 6?

17x y 7z 1 0

x 3y 2z 1 0 x 3y 2z 8 0

17x y 7z 6 0

x 1

17 y 5

1 z 3 7 .

x

1y 1

3 z 1 2 z 3 3t

y 3t x 2 4t

z 3 z 3

z 3 t y 3t y t

y t x 2 2t x 2 2t

x 2 2t

4, 3, 3? 2, 0, 3

5i j k, i 2j 3k

3i 2k, 2j k i j 2k, i j k

2i j, i k

(3)

14. Find the center and the radius of the sphere given by

(a) Center: (b) Center: (c) Center:

Radius: 2 Radius: Radius: 2

(d) Center: (e) None of these

Radius:

15. Identify the quadric surface given by

(a) Elliptic cone (b) Elliptic paraboloid (c) Hyperbolic paraboloid (d) Hyperboloid of one sheet (e) Hyperboloid of two sheets

16. Identify the quadric surface sketched at the right.

(a) Hyperboloid of two sheets (b) Elliptic paraboloid (c) Hyperboloid of one sheet (d) Elliptic cone (e) None of these

17. Find the equation of the surface of revolution if the generating curve, is revolved about the y-axis.

(a) (b)

(c) (d)

(e) None of these

18. Express the cylindrical point in spherical coordinates.

(a) (b) (c)

19. Find an equation in spherical coordinates for the surface given by the rectangular equation

(a) (b) (c)

20. Find the work done in moving a particle from if the magnitude and direction of the force are given by

(a) 5 (b) 35 (c) 17

v 5, 6, 2.P3, 1, 0 to Q2, 3, 1

tan csc sec 2

cot csc sec 2

r² z sec 2

cot csc x² y² z.

^5,2, 0.93

^5,2, 53

^5,2, 3 5

^5,2, 60

^4,2, 3

3x² y² z² 4x 1 0 4x² y² z² 4x 1 0

3x² z² 4x 1 0 4x² y² 4z² 2y 1 0

y 2x 1,

x

y z

2 2

4 4

−4

zx² 4 y²

16.

37

1, 1, 3

37

1, 1, 3

x² y² z² 2x 2y 6z 7 0.

(4)

(a) (b)

(c) (d)

(e) None of these

2. Calculate

(a) (b) (c)

3. Calculate

(a) (b) (c)

4. Calculate

(a) (b) (c)

5. Calculate

(a) (b) 0 (c)

6. Calculate the component of u in the direction of v.

(a) (b) (c)

7. Calculate is the angle between u and v.

(a) (b) (c)

239

3 226 9

239 9

226

cos where

9 26i 3

26j 6 13k

9

26

3

26 24

26 i 8j

7

4

u v^w.

1, 1, 1

3

7, 7, 7

u v.

1, 1, 1

3

7, 7, 7

u^v.

u i 2j k, v 3i j 4k and w 2j k.

i

j

k

i

j k.

1

51i 1

51j 1

51k

1

51i 1

51j 7

51k

i j k i j 7k

P2, 1, 3 to Q1, 0, 4.

Name

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Date

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Class

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Section

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Test Form B Chapter 11

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8. Which of the following vectors is orthogonal to both u and w?

(a) (b) (c) v

9. Which of the following statements is true about the vectors

(a) are orthogonal. (b) are parallel.

(c) (d) The angle between

(e) None of these

10. Which of the following is a set of parametric equations for the line through the points and

(a) (b) (c)

11. Find an equation of the plane that passes through the points and

(a) (b) (c)

12. If the equation of a cylinder is given by then

(a) The rulings are parallel to the x-axis. (b) The graph is a parabola.

(c) The rulings are parallel to the z-axis. (d) The generating curve is a parabola.

(e) None of these

13. Calculate the distance from the point to the plane given by

(a) (b) (c)

14. Find the center and radius of the sphere given by the equation

(a) Center: (b) Center: (c) Center:

Radius: 4 Radius: 4 Radius:

(d) Center: (e) None of these

Radius:10

2, 1, 1

10

2, 1, 1

x² y² z² 4x 2y 2z 10.

65

16

21 14

21

14

x 3 2y 1 4z 0.

1, 2, 3

x² z 0, 7x 4y 5z 2 0

7x 4y 5z 10 0 7x 6y 5z 20 0

2x 4y 5z 10 0

2, 1, 0. 2, 1, 4, 3, 1, 3

z 3t y 2 3t x 3 4t

z 3t z 3

z 3 3t y 3 2t y 2 5t

y 3 t x 4 3t x 3 t

x 4 7t

4, 3, 3?

3, 2, 0

u and v is 4. u is a unit vector of v.

u and v u and v

v 2i ⁴3j k? u¹2i¹3j¹4k and 4j k

w u j

(6)

15. Identify the quadric surface given by

(a) Elliptic cone (b) Elliptic paraboloid (c) Hyperbolic paraboloid (d) Hyperboloid of one sheet (e) Hyperboloid of two sheets

16. Identify the quadric surface sketched at the right.

(a) Hyperboloid of two sheets (b) Elliptic paraboloid (c) Hyperboloid of one sheet (d) Elliptic cone (e) None of these

17. Find the equation of revolution if the generating curve, is revolved about the x-axis.

(a) (b)

(c) (d)

(e) None of these

18. Express the spherical point in rectangular coordinates.

(a) (b) (c)

19. Find an equation in cylindrical coordinates for the surface given by the rectangular equation

(a) (b) (c)

20. Find the work done in moving a particle from if the magnitude and direction of the force are given by

(a) 15 (b) 21 (c) 168

v 4, 2, 3.P0, 2, 1 to Q3, 2, 2

cot csc

cot csc sec 2

r² z sec 2

r² z x² y² z.

³²^,³²³^,³²

²³²^,³²³²^,³²

³²²³^,³²³²^,³²

³²^,³²^,³²³

^3,³4, 3

3x² y² z² 4x 1 0 4x² y² z² 4x 1 0

3x² z² 4x 1 0 4x² y² 4z² 2y 1 0

y 2x 1,

x y

z

2 2 2

4 4

−4

z²x² 4 y²

16.

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1. A vector v has initial point and terminal point Find the unit vector in the direction of v.

(a) (b) (c)

2. Find a vector with magnitude in the direction of

(a) (b) (c)

3. Forces with magnitudes of 200 and 300 pounds act on a machine part at angles of respectively, with the positive x-axis. Find the direction of the resultant forces.

(a) (b) (c)

4. Find the standard equation for the sphere that has points as endpoints of a diameter.

(a) (b)

(c) (d)

(e) None of these

5. Determine which vector is parallel to the vector

(a) (b) (c)

6. Find the unit vector in the direction of the vector

(a) (b) (c)

3i1 3j 2

3k

i j k 2

9i1 9j 2

9k 2

5i 1

5j 3

5k

v 2i j 2k.

6, 9, 3

^1,³²^,¹²

²³^{, 1,}¹³

4, 6, 2

v 2, 3, 1.

x 5²y 2²z 3² 6

x 1²y 1²z 2² 36

x 1²y 1²z 2² 38

4, 3, 5 and 6, 1, 1

7.5

82.8 7.1

7.1

60 and 45,

^2,²

^5,⁵

^10,¹⁰

¹2, 1

2

v 1, 1.

10 1

5i 2j

4 5i3

5j 3

5i4 5j 1

5i 2

5j

2, 1.

1, 3

Name

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Date

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Section

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Test Form C Chapter 11

(8)

7. Find and the angle between vectors

(a) (b) (c) 300

8. Find the vector component of u orthogonal to v for

(a) (b) (c)

9. Calculate the angle that vector makes with the positive y-axis.

(a) (b) (c)

10. ______________

(a) (b) (c)

11. Calculate and

(a) (b) (c)

12. Find the area of the parallelogram having vectors as adjacent sides.

(a) 3 (b) 10 (c)

13. Find an equation of the plane determined by the points

(a) (b) (c)

14. Find the point of intersection of the line given by the parametric equations

and with the yz-plane.

(a) (b) (c)

(d) 7, 33, 7 (e) None of these

3, 7, 2

^0,³⁷³^,⁴³

^0,^5,⁷²

z 2 t x 3 2t, y 7 8t,

14x 6y 3z 17

6x 2y z 5 6x 2y z 0

18i 6j 3k 0

1, 2, 3, 2, 3, 1, and 0, 2, 1.

69

101 v₁ i 2j 2k and v2 3i 2j k 11i 11j 11k

11i 11j 11k 13i 7j 5k

11i 11j 11k

v 2i j 3k.

u v for u 3i 4j k proj_uv

v² v

v v

v^v 80.3

147.7 32.3

59.5

v 3i 5j k 1

2i5 2j

1

2i 1

2j 3

2i3 2j

1 2i 1

2j

u i 2j and v 4i 4j.

300

900

3003

u and v is 2 3. u^{v if} u 40, v 15,

(9)

15. Find a generating curve for the surface of revolution given by

(a) (b) (c)

(d) Both (a) and (b) (e) None of these

16. Write the equation in standard form and identify the quadric surface given by (a)

(b)

(c)

(d)

(e) None of these

17. Express the spherical coordinate point in cylindrical coordinates.

(a) (b) (c)

18. Find a rectangular equation for the graph represented by the spherical equation

(a) (b) (c)

(d) z 2x² (e) None of these

y² z² x² 0 x² y² z² 2z

x² y² z² 2x

2 cos .

³^3,6, 3

³^{3, 3,}3

³2³, 3

2, 3

³2, 33

2 , 3

^6,6, 3

x² 9

4y² 9z, Hyperbolic paraboloid zx²

9 y²

4, Hyperboloid of one sheet zx²

9 y²

4, Hyperbolic paraboloid 36z 4x² 9y², Cone

4x² 9y² 36z 0.

x y² z² z3x 2x²

y3x 2x²

2x² y² z² 3x.

(10)

2. Calculate

3. Calculate

4. Calculate

5. Calculate

6. Calculate

7. Calculate where is the angle between u and v.

8. Calculate a vector perpendicular to both u and w.

9. Let Show that

the vectors and parallel to lines respectively, are orthogonal.

10. Find parametric equations for the line through the point and parallel to the line given by

11. Write an equation of the plane that is determined by the lines given by and

12. Sketch the plane given by

13. Find the numbers x and y such that the point lies on the line passing through the points and

14. Write the equation of the sphere, in standard form

and find the center and radius.

15. Identify and sketch the quadric surface given by x² 1 y²

4 z 0.

4x² 4y² 4z² 4x 16y 8z 9 0,

0, 3, 2.

2, 5, 7 x, y, 1

3x 4y 2z 12.

x 4

2 y 3

1 z 1 1 .

x 6

6 y 1

2 z

3 x

1y 2

2 z 1 5 .

4, 1, 3

L₁ and L₂, v₂

v₁

z 1 2t.

L₁ : x 2 4t, y 1 5t, z 7t and L2 : x 2 t, y 5 2t,

proj_vu.

u^v^w^.

u v.

u^v.

u i 2j k, v 2i j k and w 3i k.

i k.

P2, 1, 3 to Q1, 0, 4.

Name

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Date

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Section

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Test Form D Chapter 11

(11)

16. Sketch the surface represented by the equation

17. Find the equation of the surface of revolution if the generating curve, is revolved about the x-axis.

18. Convert the point from rectangular to cylindrical coordinates.

19. Find an equation in rectangular coordinates for the spherical coordinate equation and identify the surface.

20. Find the area of the parallelogram having vectors and as adjacent sides.

v 5i 2k u i 2j 6k

9 csc csc

3, 3, 7

2x 3z 1, y x².

(12)

1. A vector v has initial point and terminal point Find the unit vector in the direction of v.

2. Find a vector with magnitude 3 in the direction of the vector

3. Three forces with magnitude of 50, 20, and 40 pounds act on an object at angles of and respectively, with the positive x-axis. Find the direction and magnitude of the resultant forces.

4. Find the standard equation for the sphere that has points as endpoints of a diameter.

5. Determine whether each vector is parallel to the vector If it is, find c such that

a. b.

c. d.

6. A vector v has initial point and terminal point a. Write v in the component form.

b. Write v as a linear combination of the standard unit vectors.

c. Find the magnitude of v.

d. Find the unit vector in the direction of v.

e. Find the unit vector in the direction opposite that of v.

7. Find and the angle between u and v is

8. Let and (a) Find the projection of u onto v.

(b) Find the vector component of u orthogonal to v.

9. Find direction cosines for the vector with initial point and terminal point

10. Determine a scalar, k, so that the vectors u 3i 2j and v 2i kjare orthogonal.

4, 3, 5.

2, 1, 3

v 3i 2j k.

u i 2j 2k

4. u^{v if}u 7, v 12,

4, 2, 1.

2, 1, 3

u 2i ¹2j⁵2k u 3i ³4j¹⁵4k

u 8i 4j 10k u 8i 4j 10k

v cu. v 4i j 5k.

2, 5, 1 and 3, 7, 3

90, 60, 30,

v 1, 2.

2, 3.

2, 1

Name

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Section

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Test Form E Chapter 11

(13)

11. Let a. Find

b. Show that the vector is orthogonal to v.

12. Calculate the area of the triangle with vertices

13. Find parametric equations for the line perpendicular to the lines given by

and passing through the point

14. Find the point of intersection of the line and the plane

15. Sketch the surface represented by

16. Identify each of the quadric surfaces:

a.

b.

c.

17. Consider the surface whose rectangular equation is

a. Find an equation for the surface in cylindrical coordinates.

b. Find an equation for the surface in spherical coordinates.

18. Sketch the surface represented by the spherical equation 2 csc .

x² y² z² 1.

2x² y² z² 1 2x² 5z² y²

3x² 3y² 3z² 2x 3y 11 0 y x² z². 2x 3y z 3 0.

x 2

2 y 3

3 z 1 1

2, 4, 1.

x 3

7 y 2

2 z 8 3

x 2

4 y 3

7 z 1 3

3, 1, 2, 2, 1, 5, and 1, 2, 2.

u v u v.

u i 2j 2k and v 3i 2j k.

Name Class. Date Section. Test Form A Chapter 11. Chapter 11 Test Bank 155

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Test Form A Chapter 11

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Test Form B Chapter 11

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Test Form C Chapter 11

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Test Form D Chapter 11

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Test Form E Chapter 11