Solution
!
!
Draw a sketch and determine 冨u v 冨.
Using the cosine law,
!
!
!
!
! !
冨u v 冨2 冨u 冨2 冨v 冨2 2冨u 冨冨v 冨 cos u
!
!
u 1v
冨u v 冨2 42 52 214 2 15 2 cos 60°
608
!
!
冨u v 冨2 21
1208 u
v
!
!
冨u v 冨 V21
!
!
To create a unit vector in the same direction as u v , multiply by the scalar equal
!
!
!
!
1
1
! . In this case, the unit vector is
1u v 2 1 u 1 v .
to 0 !
0
uv
兹21
兹21
兹21
IN SUMMARY
Key Idea
!
!
• For the vector ka where k is a scalar and a is a nonzero vector:
!
!
!
• If k 7 0, then ka is in the same direction as a with magnitude k冨a 冨.
!
!
!
• If k 6 0, then ka is in the opposite direction as a with magnitude 冨k冨冨a 冨.
Need to Know
• If two or more vectors are nonzero scalar multiples of the same vector, then
all these vectors are collinear.
1 !
• ! x is a vector of length one, called a unit vector, in the direction of the
冨x冨
!
nonzero vector x .
!
1 !
• ! x is a unit vector in the opposite direction of the nonzero vector x .
冨x冨
Exercise 6.3
PART A
!
!
1. Explain why the statement a 2 @ b @ is not meaningful.
298 6 . 3
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2. An airplane is flying at an airspeed of 300 km>h. Using a scale of 1 cm
equivalent to 50 km>h, draw a velocity vector to represent each of the following:
a. a speed of 150 km>h heading in the direction N45°E
b. a speed of 450 km>h heading in the direction E15°S
c. a speed of 100 km>h heading in an easterly direction
d. a speed of 300 km/h heading on a bearing of 345°
3. An airplane’s direction is E25°N. Explain why this is the same as N65°E or
a bearing of 65°.
!
!
4. The vector v has magnitude 2, i.e., 冨v 冨 2. Draw the following vectors
!
and express each of them as a scalar multiple of v .
!
a. a vector in the same direction as v with twice its magnitude
!
b. a vector in the same direction as v with one-half its magnitude
!
c. a vector in the opposite direction as v with two-thirds its magnitude
!
d. a vector in the opposite direction as v with twice its magnitude
!
e. a unit vector in the same direction as v
v
PART B
K
!
!
5. The vectors x and y are shown below. Draw a diagram for each
of the following.
y
x
!
!
a. x 3y
!
!
!
!
!
!
b. x 3y
c. 2x y
d. 2x y
!
!
6. Draw two vectors, a and b , that do not have the same magnitude and are
noncollinear. Using the vectors you drew, construct the following:
!
!
!
!
!
!
!
a. 2a
b. 3b
c. 3b
d. 2a 3b
e. 2a 3b
! !
!
! 2 !
! 1 !
7. Three collinear vectors, a , b , and c , are such that a 3 b and a 2 c .
!
!
!
a. Determine integer values for m and n such that mc nb 0 . How many
values are possible for m and n to make this statement true?
!
!
!
!
b. Determine integer values for d, e, and f such that da eb f c 0 . Are
these values unique?
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CHAPTER 6
299
!
!
!
!
8. The two vectors a and b are collinear and are chosen such that 冨a 冨 @ b @ .
Draw a diagram showing different possible configurations for these
two vectors.
!
!
!
!
9. The vectors a and b are perpendicular. Are the vectors 4a and 2b also
perpendicular? Illustrate your answer with a sketch.
!
!
10. If the vectors a and b are noncollinear, determine which of the following
pairs of vectors are collinear and which are not.
!
! !
!
!
! !
!
a. 2a , 3a
b. 2a , 3b
c. 5a , 32 b
d. b , 2b
C
!
11. In the discussion, we defined 1! x . Using your own scale, draw your own
!
冨x冨
vector to represent x .
!
a. Sketch 1! x and describe this vector in your own words.
冨x 冨
!
b. Sketch 1! x and describe this vector in your own words.
冨x冨
!
!
!
!
12. Two vectors, a and b , are such that 2a 3b . Draw
a possible sketch
!
!
of these two vectors. What is the value of m, if @ b @ m冨a 冨?
A
13. The points B, C, and! D are drawn on line segment AE dividing it into four
!
!
!
equal lengths. If AD a , write each of the following in terms of a and 冨a 冨.
A
B
C
D
E
AD = a
!
a. EC
!
b. BC
!
c. @ ED @
!
d. @ AC @
!
e. AE
!
!
14. The vectors x and y are unit vectors that make an angle of 90° with each
!
!
!
!
other. Calculate the value of 冨2x y 冨 and the direction of 2x y .
!
!
15. The vectors x and y are unit vectors that make an angle of 30° with each
!
!
!
!
other. Calculate the value of 冨2x y 冨 and the direction of 2x y .
!
!
16. Prove that 1! a is a unit vector pointing in the same direction as a .
冨a冨
!
!
(Hint: Let b 1! a and then find the magnitude of each side
冨a 冨
of this equation.)
T
300 6 . 3
17. In ^ABC, a median is drawn from vertex A to the midpoint of BC,
!
!
!
! 1 ! 1 !
!
which is labelled D. If AB b and AC c , prove that AD 2 b 2 c .
M U LT I P L I C AT I O N O F A V E C TO R B Y A S C A L A R
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18. Let PQR be a triangle !in which M is! the !midpoint of PQ and N is the
!
!
midpoint
of
PR.
If
and
find
vector
expressions
for
PM
a
PN
b
,
MN
!
!
!
and QR in terms of a and b . What conclusions can be drawn about MN and
QR? Explain.
P
N
M
Q
R
19. Draw rhombus ABCD where AB 3 cm. For each of the following, name two
!
!
vectors u and v in your diagram such that
!
!
!
!
a. u v
c. u v
!
!
!
!
b. u 2v
d. u 0.5v
PART C
!
!
!
!
20. Two vectors, x! and y , are drawn such that 冨x 冨 3冨y 冨. Considering
!
!
mx ny 0 , determine all possible values for m and n such that
!
!
a. x and y are collinear
!
!
b. x and y are noncollinear
!
!
!
!
21. ABCDEF is a regular hexagon such that AB a and BC b .
!
!
!
a. Express CD in terms of a and b .
!
!
b. Prove that BE is parallel to CD and that @ BE @ 2 @ CD @ .
A
F
B
E
C
D
22. ABCD is a trapezoid whose diagonals AC and BD intersect at the point E.
! 2 !
! 3 ! 2 !
If AB 3 DC , prove that AE 5 AB 5 AD .
A
D
B
E
C
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