9781118230725.pdf

56 CHAPTER 3 VECTORS

10 Figure 3-25 shows vector and

four other vectors that have the same

magnitude but differ in orientation.

(a) Which of those other four vectors

have the same dot product with? (b)

Which have a negative dot product

with?

11 In a game held within a three-

dimensional maze, you must move

your game piece from start, at xyzco-

ordinates (0, 0, 0), to finish, 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 the

game. Which displacements and in what sequence will get your

game piece to finish?

 7  2  3  2  3  2

 2  4  3  5 3.

12 The xandycomponents of four vectors , , , and are given

below. 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 dy3.

13 Which of the following are correct (meaningful) vector

expressions? 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

:

:a b::c :d

:q ˆi ˆj kˆ :s ˆi ˆj kˆ

:p ˆi ˆj kˆ :r ˆi ˆj kˆ

A

:

A

:

A

:

B

A

C

E

D

θ

θ

θ

θ

Figure 3-25Question 10.

F F

v F

x v x vx

z z z

y y y

(1) (2) (3)

Figure 3-24Question 9.

Figure 3-23Question 5.

Questions

1 Can the sum of the magnitudes
of two vectors ever be equal to the
magnitude of the sum of the same
two vectors? If no, why not? If yes,
when?

2 The two vectors shown in Fig. 3-21
lie in an xyplane. What are the signs
of the xandycomponents, respec-
tively, of (a) , (b) , and
(c)?

3 Being part of the “Gators,” the
University of Florida golfing team
must play on a putting green with an
alligator pit. Figure 3-22 shows an
overhead view of one putting chal-
lenge of the team; an xycoordinate
system is superimposed. Team mem-
bers must putt from the origin to the
hole, which is at xycoordinates (8 m,
12 m), but they can putt the golf ball
using only one or more of the fol-
lowing displacements, one or more
times:

,.

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

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

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

b :a
b
:
:a

b

:

a:

d 3

:

d 2 (8 m)iˆ

:

d 1 (6 m)ˆj,

:

(8 m)iˆ(6 m)jˆ

d: 2 d: 1

d: 1 d: 2 d: 1 d: 2

6 Describe two vectors and such that

(a)   and abc;

(b)    ;

(c)   and a^2 b^2 c^2.

7 If   ( ), does (a)  ( )  ( ), (b) 

( )  , and (c)  ( ) ?

8 If    , must equal?

9 If q(  ) and is perpendicular to , then what is the

direction of in the three situations shown in Fig. 3-24 when con-

stantqis (a) positive and (b) negative?

B:

:F :v B: :v B:

:a :b :a :c :b :c

:b :d :c :c :d :a :b

:d :a :b :c a: :d :c :b :a

:a b: :c

:a b: :a b:

:a :b :c

b

:

y :a

x

d 2

d 1

Figure 3-21Question 2.

Hole

Gator

pit

y

x

Figure 3-22Question 3.

z

y

x

(a )

z y

x

(b )

x

z

y

(c )

y y

y

x

z

(d )

x

z

(e )

z

x

( f )