CHAPTE R 3 VECTORS
52
If and are parallel or antiparallel, ϫ ϭ0. The magnitude of ϫ , which can
be written as
, is maximum when and are perpendicular to each other.
b
:
a
:
͉a
: ϫ b
: ͉
b
:
a
:
b
:
a
:
b
:
a
:
where f is the smaller of the two angles between and . (You must use the
b
:
a
:
The direction of
is perpendicular to the plane that contains
and .
b
:
a
:
c
:
Checkpoint 4
Vectors and have magnitudes of 3 units and 4 units, respectively. What is the
angle between the directions of and if
equals (a) zero, (b) 12 units, and
(c) 12 units?
Ϫ
D
:
C
: ؒ
D
:
C
:
D
:
C
:
The Vector Product
The vector product of and , written ϫ , produces a third vector whose
magnitude is
c ϭ ab sin f,
(3-24)
c
:
b
:
a
:
b
:
a
:
them is 90°.) Also, we used the right-hand rule to get the direction of ϫ as
being in the positive direction of the z axis (thus in the direction of ).
k ˆ
j
ˆ
i
ˆ
smaller of the two angles between the vectors because sin f and sin(360° Ϫ f)
differ in algebraic sign.) Because of the notation, ϫ is also known as the cross
product, and in speech it is “a cross b.”
b
:
a
:
Figure 3-19a shows how to determine the direction of ϭ ϫ with what is
known as a right-hand rule. Place the vectors and tail to tail without altering
their orientations, and imagine a line that is perpendicular to their plane where
they meet. Pretend to place your right hand around that line in such a way that
your fingers would sweep into through the smaller angle between them. Your
outstretched thumb points in the direction of .
The order of the vector multiplication is important. In Fig. 3-19b, we are
determining the direction of
, so the fingers are placed to sweep
into through the smaller angle. The thumb ends up in the opposite direction
from previously, and so it must be that
; that is,
.
(3-25)
In other words, the commutative law does not apply to a vector product.
In unit-vector notation, we write
ϫ ϭ (a x ϩ a y ϩ a z ) ϫ (b x ϩ b y ϩ b z ),
(3-26)
which can be expanded according to the distributive law; that is, each component
of the first vector is to be crossed with each component of the second vector. The
cross products of unit vectors are given in Appendix E (see “Products of
Vectors”). For example, in the expansion of Eq. 3-26, we have
a x ϫ b x ϭ a x b x ( ϫ ) ϭ 0,
because the two unit vectors and are parallel and thus have a zero cross product. Similarly, we have
a x ϫ b y ϭ a x b y ( ϫ ) ϭ a x b y .
In the last step we used Eq. 3-24 to evaluate the magnitude of ϫ as unity.
(These vectors and each have a magnitude of unity, and the angle between
j
ˆ
i
ˆ
j
ˆ
i
ˆ
k ˆ
j
ˆ
i
ˆ
j
ˆ
i
ˆ
i
ˆ
i
ˆ
i
ˆ
i
ˆ
i
ˆ
i
ˆ
k ˆ
j
ˆ
i
ˆ
k ˆ
j
ˆ
i
ˆ
b
:
a
:
b
: ϫ a
: ϭ Ϫ(a
: ϫ b
: )
cЈ
: ϭ Ϫc
:
a
:
b
:
cЈ
: ϭ b
: ϫ a
:
c
:
b
:
a
:
b
:
a
:
b
:
a
:
c
:
52
If and are parallel or antiparallel, ϫ ϭ0. The magnitude of ϫ , which can
be written as
, is maximum when and are perpendicular to each other.
b
:
a
:
͉a
: ϫ b
: ͉
b
:
a
:
b
:
a
:
b
:
a
:
where f is the smaller of the two angles between and . (You must use the
b
:
a
:
The direction of
is perpendicular to the plane that contains
and .
b
:
a
:
c
:
Checkpoint 4
Vectors and have magnitudes of 3 units and 4 units, respectively. What is the
angle between the directions of and if
equals (a) zero, (b) 12 units, and
(c) 12 units?
Ϫ
D
:
C
: ؒ
D
:
C
:
D
:
C
:
The Vector Product
The vector product of and , written ϫ , produces a third vector whose
magnitude is
c ϭ ab sin f,
(3-24)
c
:
b
:
a
:
b
:
a
:
them is 90°.) Also, we used the right-hand rule to get the direction of ϫ as
being in the positive direction of the z axis (thus in the direction of ).
k ˆ
j
ˆ
i
ˆ
smaller of the two angles between the vectors because sin f and sin(360° Ϫ f)
differ in algebraic sign.) Because of the notation, ϫ is also known as the cross
product, and in speech it is “a cross b.”
b
:
a
:
Figure 3-19a shows how to determine the direction of ϭ ϫ with what is
known as a right-hand rule. Place the vectors and tail to tail without altering
their orientations, and imagine a line that is perpendicular to their plane where
they meet. Pretend to place your right hand around that line in such a way that
your fingers would sweep into through the smaller angle between them. Your
outstretched thumb points in the direction of .
The order of the vector multiplication is important. In Fig. 3-19b, we are
determining the direction of
, so the fingers are placed to sweep
into through the smaller angle. The thumb ends up in the opposite direction
from previously, and so it must be that
; that is,
.
(3-25)
In other words, the commutative law does not apply to a vector product.
In unit-vector notation, we write
ϫ ϭ (a x ϩ a y ϩ a z ) ϫ (b x ϩ b y ϩ b z ),
(3-26)
which can be expanded according to the distributive law; that is, each component
of the first vector is to be crossed with each component of the second vector. The
cross products of unit vectors are given in Appendix E (see “Products of
Vectors”). For example, in the expansion of Eq. 3-26, we have
a x ϫ b x ϭ a x b x ( ϫ ) ϭ 0,
because the two unit vectors and are parallel and thus have a zero cross product. Similarly, we have
a x ϫ b y ϭ a x b y ( ϫ ) ϭ a x b y .
In the last step we used Eq. 3-24 to evaluate the magnitude of ϫ as unity.
(These vectors and each have a magnitude of unity, and the angle between
j
ˆ
i
ˆ
j
ˆ
i
ˆ
k ˆ
j
ˆ
i
ˆ
j
ˆ
i
ˆ
i
ˆ
i
ˆ
i
ˆ
i
ˆ
i
ˆ
i
ˆ
k ˆ
j
ˆ
i
ˆ
k ˆ
j
ˆ
i
ˆ
b
:
a
:
b
: ϫ a
: ϭ Ϫ(a
: ϫ b
: )
cЈ
: ϭ Ϫc
:
a
:
b
:
cЈ
: ϭ b
: ϫ a
:
c
:
b
:
a
:
b
:
a
:
b
:
a
:
c
: