Cross product
1
Chapter 7
Cross product
We are getting ready to study
integration
in several variables.
Until now we have been
doing only
differential
calculus.
One outcome of this study will be our ability to compute
volumes of interesting regions of
R
n
.
As preparation for this we shall learn in this chapter
how to compute volumes of parallelepipeds in
R
3
. In this material there is a close connection
with the vector product, which we now discuss.
A.
Definition of the cross product
We begin with a simple but interesting problem.
Let
x
,
y
be given vectors in
R
3
:
x
=
(
x
1
, x
2
, x
3
) and
y
= (
y
1
, y
2
, y
3
). Assume that
x
and
y
are linearly independent; in other words,
0,
x
,
y
determine a unique plane.
We then want to determine a nonzero vector
z
which is
orthogonal to this plane. That is, we want to solve the equations
x
•
z
= 0 and
y
•
z
= 0. We
are certain in advance that
z
will be uniquely determined up to a scalar factor. The equations
in terms of the coordinates of
z
are
x
1
z
1
+
x
2
z
2
+
x
3
z
3
=
0
,
y
1
z
1
+
y
2
z
2
+
y
3
z
3
=
0
.
It is no surprise that we have two equations but three “unknowns,” as we know
z
is not going
to be unique. Since
x
and
y
are linearly independent, the matrix
x
1
x
2
x
3
y
1
y
2
y
3
¶
has row rank equal to 2, and thus also has column rank 2. Thus it has two linearly independent
columns. To be definite, suppose that the first two columns are independent; in other words,
x
1
y
2

x
2
y
1
6
= 0
.
(This is all a special case of the general discussion in Section 6B.) Then we can solve the
following two equations for the “unknowns”
z
1
and
z
2
:
x
1
z
1
+
x
2
z
2
=

x
3
z
3
,
y
1
z
1
+
y
2
z
2
=

y
3
z
3
.
The result is
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Chapter 7
z
1
=
x
2
y
3

x
3
y
2
x
1
y
2

x
2
y
1
z
3
,
z
2
=
x
3
y
1

x
1
y
3
x
1
y
2

x
2
y
1
z
3
.
Notice of course we have an undetermined scalar factor
z
3
.
Now we simply make the choice
z
3
=
x
1
y
2

x
2
y
1
(
6
= 0). Then the vector
z
can be written
z
1
=
x
2
y
3

x
3
y
2
,
z
2
=
x
3
y
1

x
1
y
3
,
z
3
=
x
1
y
2

x
2
y
1
.
This is precisely what we were trying to find, and we now simply make this a definition:
DEFINITION.
The
cross product
(or
vector product
) of two vectors
x
,
y
in
R
3
is the vector
x
×
y
= (
x
2
y
3

x
3
y
2
, x
3
y
1

x
1
y
3
, x
1
y
2

x
2
y
1
)
.
DISCUSSION.
1. Our development was based on the assumption that
x
and
y
are linearly
independent.
But the
definition
still holds in the case of linear dependence, and produces
x
×
y
= 0. Thus we can say immediately that
x
and
y
are linearly dependent
⇐⇒
x
×
y
= 0
.
2.
We also made the working assumption that
x
1
y
2

x
2
y
1
6
= 0.
Either of the other two
choices of independent columns produces the same sort of result. This is clearly seen in the
nice symmetry of the definition.
3. The definition is actually quite easily memorized. Just realize that the first component of
z
=
x
×
y
is
z
1
=
x
2
y
3

x
3
y
2
and then a cyclic permutation 1
→
2
→
3
→
1 of the indices automatically produces the other
two components.
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 Fall '08
 Hatcher
 Calculus, Dot Product

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