7/25/2019 Vectors Problems.pdf

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CHAPTER 3 VECTORS56

10 Figure 3-25 shows vector andfour other vectors that have the samemagnitude but differ in orientation.(a) Which of those other four vectorshave the same dot product with ? (b)Which have a negative dot productwith ?

11 In a game held within a three-dimensional maze, you must moveyour game piece from start, atxyz co-

ordinates (0, 0, 0), tofinish, at coordinates (2 cm, 4 cm, 4 cm).The game piece can undergo only the displacements (in centime-ters) given below. If, along the way, the game piece lands at coordi-nates (5 cm, 1 cm, 1 cm) or (5 cm, 2 cm, 1 cm), you lose thegame. Which displacements and in what sequence will get yourgame piece tofinish?

7 2 3 2 3 2

2 4 3 5 3 .

12 Thex andy components of four vectors , , , and are givenbelow.For which vectors will your calculator give you the correct an-gleuwhen you use it to find uwith Eq. 3-6? Answer first by examin-ing Fig.3-12,and then check your answers with your calculator.

ax 3 ay 3 cx 3 cy 3

bx 3 by 3 dx 3 dy 3.13 Which of the following are correct (meaningful) vectorexpressions? What is wrong with any incorrect expression?

(a) ( ) (f) ( )

(b) ( ) (g) 5

(c) ( ) (h) 5 ( )

(d) ( ) (i) 5 ( )

(e) ( ) (j) ( ) ( )C:

B:

B:

A:

C:

B:

A:

C:

B:

C:

B:

A:

C:

B:

C:

B:

A:

A:

C:

B:

A:

C:

B:

A:

C:

B:

A:

d:

c:b:

a:

kjis:kjiq:kjir:kjip:

A:

A:

A:

B

A

C

E

D

Figure 3-25 Question 10.

F F

Fv

v v

xxx

z z z

yyy

(1) (2) (3)

Figure 3-24 Question 9.

Figure 3-23 Question 5.

Questions

1 Can the sum of the magnitudesof two vectors ever be equal to themagnitude of the sum of the same

two vectors? If no, why not? If yes,when?

2 The two vectors shown in Fig.3-21lie in an xy plane. What are the signsof the x and y components, respec-tively, of (a) , (b) , and(c) ?

3 Being part of the Gators, theUniversity of Florida golfing teammust play on a putting green with analligator pit. Figure 3-22 shows anoverhead view of one putting chal-lenge of the team; an xy coordinatesystem is superimposed. Team mem-bers must putt from the origin to thehole, which is atxy coordinates (8 m,12 m), but they can putt the golf ballusing only one or more of the fol-lowing displacements, one or moretimes:

, .

The pit is at coordinates (8 m, 6 m). If a team member putts theball into or through the pit, the member is automatically trans-ferred to Florida State University, the arch rival.What sequenceof displacements should a team member use to avoid the pit andthe school transfer?

4 Equation 3-2 shows that the addition of two vectors and iscommutative. Does that mean subtraction is commutative, so that

?5 Which of the arrangements of axes in Fig. 3-23 can be labeledright-handed coordinate system? As usual, each axis label indi-cates the positive side of the axis.

a:

b

:

b

:

a:

b:

a:

d3:

(8 m)id2:

(6 m)j,d1

:

(8 m)i (6 m)j

d2:

d1:

d1:

d2:

d1:

d2:

6 Describe two vectors and such that

(a) and a b c;

(b)

;(c) and a2 b2 c2.

7 If ( ), does (a) ( ) ( ), (b) ( ) , and (c) ( ) ?

8 If , must equal ?

9 If q( ) and is perpendicular to , then what is thedirection of in the three situations shown in Fig. 3-24 when con-stantq is (a) positive and (b) negative?

B:

B:

v:B:

v:F:

c:b:

c:a:b:

a:

b:

a:d:

c:c:d:

b:

a:b:

c:d:

a:c:b:

a:d:

c:b:

a:b

:

a

:

b

:

a

:

c:b:

a:

b:

a:y

x

d2

d1

Figure 3-21 Question 2.

Hole

Gatorpit

x

Figure 3-22 Question 3.

z

y

x

(a)

yz

x

(b)

x

z

y

(c)

yy

y

x

z

(d)

x

z

(e)

z

x

( f)