Florida Gulf Coast University - PHY 2048

Chapter 3—Vectors

MULTIPLE CHOICE

Instructions: On occasion, the notation             = [A, ?] will be a shorthand notation for

.

1)If       = [15, 80?] and                              , what is the magnitude of      ?

1. 1. 15
   2. 35
   3. 32

d. 5.0

e.   23

2. Vectors   and    are shown. What is the magnitude of a vector                if  ?

3. If                      and                       , what is the magnitude of the vector            ?
   1. 42
   2. 22
   3. 64
   4. 90
   5. 13

4. If                      and                       , what is the direction of the vector             ?

a.   ?49?

b. ?41?

c.   ?90?

d. +49?

e.   +21?

5. If     = [10 m, 30?] and      = [25 m, 130?], what is the magnitude of the sum of these two vectors?
   1. 20 m
   2. 35 m
   3. 15 m
   4. 25 m
   5. 50 m

6. If     = [10 m, 30?] and      = [25 m, 130?], what is the direction of the sum of these two vectors?

a.   17?

b. 73?

c.   107?

d. 163?

e.   100?

7. A vector,   , when added to the vector              yields a resultant vector which is in the positive y

direction and has a magnitude equal to that of                                      . What is the magnitude of                                        ? a.                                      3.2

b.   6.3

c.   9.5

4. 18
5. 5

8. If vector    is added to vector , the result is       . If     is subtracted from , the result is           . What is the magnitude of         ?

a.   5.1

b.   4.1

c.   5.4

d.   5.8

e.   8.2

9. If     = [2.5 cm, 80?], i.e., the magnitude and direction of                                are 2.5 cm and 80?,       = [3.5 cm, 120?], and                                                   , what is the direction of     (to the nearest degree)?

a.   247?

b. 235?

c.   243?

d. 216?

e.   144?

10. If vector    is added to vector , the result is           . If     is subtracted from , the result is         .

What is the direction of (to the nearest degree)?

a.   225?

b. 221?

c.   230?

d. 236?

e.   206?

11. A vector    is added to                . The resultant vector is in the positive x direction and has a magnitude equal to                   . What is the magnitude of  ?
    1. 11

b.   5.1

c.   7.1

d.   8.3

e.   12.2

12. A vector    is added to                . The resultant vector is in the positive x direction and has a magnitude equal to that of       . What is the direction of     ?

a.   74?

b. 100?

c.   ?81?

d. ?62?

e.   106?

13. If two collinear vectors  and    are added, the resultant has a magnitude equal to 4.0. If                               is subtracted from                                                                            , the resultant has a magnitude equal to 8.0. What is the magnitude of                          ?

a.   2.0

b.   3.0

c.   4.0

d.   5.0

e.   6.0

14. If two collinear vectors  and    are added, the resultant has a magnitude equal to 4.0. If                               is subtracted from                                                                            , the resultant has a magnitude equal to 8.0. What is the magnitude of                          ?

a.   2.0

b.   3.0

c.   4.0

d.   5.0

e.   6.0

15. When vector   is added to vector , which has a magnitude of 5.0, the vector representing their sum is perpendicular to and has a magnitude that is twice that of                                       . What is the magnitude of  ?

a.   2.2

b.   2.5

c.   4.5

d.   5.0

e.   7.0

16. Starting from one oasis, a camel walks 25 km in a direction 30? south of west and then walks 30 km toward the north to a second oasis. What distance separates the two oases?
    1. 15 km
    2. 48 km
    3. 28 km
    4. 53 km
    5. 55 km

17. Starting from one oasis, a camel walks 25 km in a direction 30? south of west and then walks 30 km toward the north to a second oasis. What is the direction from the first oasis to the second oasis?
    1. 21? N of W
    2. 39? W of N
    3. 69? N of W
    4. 51? W of N
    5. 42? W of N

Instructions: On occasion, the notation = [A, ?] will be a shorthand notation for

.

Exhibit 3-1

The three forces shown act on a particle.

Use this exhibit to answer the following question(s).

18. Refer to Exhibit 3-1. What is the magnitude of the resultant of these three forces? a.          27.0 N

b.   33.2 N

c.   36.3 N

d.   23.8 N

e.   105 N

19. Refer to Exhibit 3-1. What is the direction of the resultant of these three forces?

a.   35?

b. 45?

c.   65?

d. 55?

e.   85?

Instructions: On occasion, the notation = [A, ?] will be a shorthand notation for

.

20. If vector    is added to vector , the result is a third vector that is perpendicular to  and has a magnitude equal to 3 . What is the ratio of the magnitude of           to that of      ?

a.   1.8

b.   2.2

c.   3.2

d.   1.3

e.   1.6

Instructions: On occasion, the notation = [A, ?] will be a shorthand notation for

.

Exhibit 3-2

A child starts at one corner of a cubical jungle gym in a playground and climbs up to the diagonally opposite corner. The original corner is the coordinate origin, and the x, y and z axes are oriented along the jungle gym edges. The length of each side is 2 m.

Use this exhibit to answer the following question(s).

21. Refer to Exhibit 3-2. The child's displacement is: a.

b.

c.

d.

e.

22. Refer to Exhibit 3-2. What is the child’s distance from her starting position?
    1. 2.8 m
    2. 3.5 m
    3. 6.0 m
    4. 6.9 m

e.   12.0 m

Instructions: On occasion, the notation = [A, ?] will be a shorthand notation for

.

23. The displacement of the tip of the 10 cm long minute hand of a clock between 12:15 A.M. and 12:45

P.M. is:

a.   10 cm, 90?

b. 10 cm, 180?

c.   10 cm, 4 500?

d. 20 cm, 180?

e.   20 cm, 540?

24. A student decides to spend spring break by driving 50 miles due east, then 50 miles 30 degrees south of east, then 50 miles 30 degrees south of that direction, and to continue to drive 50 miles deviating by 30 degrees each time until he returns to his original position. How far will he drive, and how many vectors must he sum to calculate his displacement?

a.   0, 0

25. Given that                              and                            , what is   ? a.

b.

c.

d.

e.

26. Given that                            and                           , what is   ? a.

b.

c.

d.

e.

27. Given that                            and                           , what is   ? a.

b.

c.

d.

e.

28. The diagram below shows 3 vectors which sum to zero, all of equal length. Which statement below is true?

29. Which statement is true about the unit vectors , and   ?
    1. Their directions are defined by a left-handed coordinate system.
    2. The angle between any two is 90 degrees.
    3. Each has a length of 1 m.
    4. If   is directed east and  is directed south,          points up out of the surface.
    5. All of the above.

30. Vectors   and    have equal magnitudes. Which statement is always true?
    1. .
    2. .
    3. is perpendicular to       .
    4. is perpendicular to       .
    5. The magnitude of                               equals the magnitude of     .

31. When three vectors,     ,           , and                                                                      are placed head to tail, the vector sum  . If the vectors all have the same magnitude, the angle between the directions of any two adjacent vectors is

a.   30?

b. 60?

c.   90?

d. 120?

e.   150?

Instructions: On occasion, the notation = [A, ?] will be a shorthand notation for

.

Exhibit 3-3

The vectors  ,    , and    are shown below.

Use this exhibit to answer the following question(s).

32. Refer to Exhibit 3-3. Which diagram below correctly represents       ?
    1. d.

1. 2. e.

c.

33. Refer to Exhibit 3-3. Which diagram below correctly represents        ?
    1. d.

1. 2. e.

c.

Instructions: On occasion, the notation = [A, ?] will be a shorthand notation for

.

Exhibit 3-4

The diagram below shows the path taken by a sailboat tacking sideways because it cannot sail directly into the wind.

Use this exhibit to answer the following question(s).

34. Refer to Exhibit 3-4. The total distance it travels is a. 1 000 m.

b.   1 732 m.

c.   2 000 m.

d.   6 298 m.

e.   8 000 m.

35. Refer to Exhibit 3-4. The total displacement of the sailboat, the vector sum of its displacements OB,

BC, CD and DE, is

1. 1. 1 732 m, East.
   2. 2 000 m, Northeast.
   3. 6 298 m, East.
   4. 8 000 m, Southeast.
   5. 8 000 m, East.

Instructions: On occasion, the notation = [A, ?] will be a shorthand notation for

.

36. Dana says any vector  can be represented as the sum of two vectors:        . Ardis says any

vector    can be represented as the difference of two vectors:       . Which one, if either, is correct?

1. 1. They are both wrong: every vector is unique.
   2. Dana is correct: Any vector can be represented as a sum of components and not as a difference.
   3. Ardis is correct: Any vector can be represented as a difference of vector components and

not as a sum.

d.

They are both correct: A difference of vectors is a sum              .

e.   They are both wrong: Vectors can be moved as long as they keep the same magnitude and direction.

37. The vector   has components +5 and +7 along the x and y axes respectively. Along a set of axes rotated 90 degrees counterclockwise relative to the original axes, the vector's components are

a.   ?7; ?5.

b. 7; ?5.

c.   ?7; 5.

d.   7; 5.

e.   7; 0.

38. Anthony has added the vectors listed below and gotten the result               . What errors has he made?

1. 1. He lost the minus sign in vector    .
   2. He read the   in     as     .
   3. He lost the minus sign in vector    .
   4. All of the above are correct.
   5. Only (a) and (b) above are correct.

39. Given the statement that                    , what can we conclude?
    1. and          .
    2. .

1. 3. and            .
   4. Any one of the answers above is correct.
   5. Only (a) and (b) may be correct.

40. Adding vectors   and    by the graphical method gives the same result for   +        and        +          . If both additions are done graphically from the same origin, the resultant is the vector that goes from the tail of the first vector to the tip of the second vector, i.e, it is represented by a diagonal of the parallelogram formed by showing both additions in the same figure. Note that a parallelogram has 2 diagonals. Keara

says that the sum of two vectors by the parallelogram method is                                         . Shamu says it is                                                                                           . Both used the parallelogram method, but one used the wrong diagonal. Which one of the vector pairs below contains the original two vectors?

a.   ;

b. ;

c.   ;

d. ;

e.   ;

41. Given two non-zero vectors,  and   , such that                          , the sum         satisfies a.

.

b.

.

c.

.

d.

.

e.

.

42. The vector    has components +5 and +7 along the x and y axes respectively. If the vector is now rotated 90 degrees counterclockwise relative to the original axes, the vector's components are now a.           ?7; ?5.

b. 7; ?5.

c.   ?7; 5.

d.   7; 5.

e.   7; 0.

43. The rectangular coordinates of a point are (5.00, y) and the polar coordinates of this point are (r, 67.4°). What is the value of the polar coordinate r in this case?

a.   1.92

b.   4.62