352
C H A P T E R
13
VECTOR GEOMETRY
(ET CHAPTER 12)
13.4 The Cross Product
(ET Section 12.4)
Preliminary Questions
1.
What is the
(
1
,
3
)
minor of the matrix
3
4
2

5

1
1
4
0
3
?
SOLUTION
The
(
1
,
3
)
minor is obtained by crossing out the first row and third column of the matrix. That is,
3
4
2

5

1
1
4
0
3
⇒

5

1
4
0
2.
The angle between two unit vectors
e
and
f
is
π
6
. What is the length of
e
×
f
?
SOLUTION
We use the Formula for the Length of the Cross Product:
e
×
f
=
e
f
sin
θ
Since
e
and
f
are unit vectors,
e
=
f
=
1. Also
θ
=
π
6
, therefore,
e
×
f
=
1
·
1
·
sin
π
6
=
1
2
The length of
e
×
f
is
1
2
.
3.
What is
u
×
w
, assuming that
w
×
u
=
2
,
2
,
1 ?
SOLUTION
By anticommutativity of the cross product, we have
u
×
w
=

w
×
u
=

2
,
2
,
1
=

2
,

2
,

1
4.
Find the cross product without using the formula:
(a)
4
,
8
,
2
×
4
,
8
,
2
(b)
4
,
8
,
2
×
2
,
4
,
1
SOLUTION
By properties of the cross product, the cross product of parallel vectors is the zero vector. In particular, the
cross product of a vector with itself is the zero vector. Since 4
,
8
,
2
=
2 2
,
4
,
1 , the vectors 4
,
8
,
2 and 2
,
4
,
1 are
parallel. We conclude that
4
,
8
,
2
×
4
,
8
,
2
=
0
and
4
,
8
,
2
×
2
,
4
,
1
=
0
.
5.
What are
i
×
j
and
i
×
k
?
SOLUTION
The cross product
i
×
j
and
i
×
k
are determined by the righthand rule. We can also use the following
figure to determine these crossproducts:
j
i
k
We get
i
×
j
=
k
and
i
×
k
=

j
6.
When is the cross product
v
×
w
equal to zero?
SOLUTION
The cross product
v
×
w
is equal to zero if one of the vectors
v
or
w
(or both) is the zero vector, or if
v
and
w
are parallel vectors.