UNIT 6
CROSS PRODUCT IN
R
3
INTRO: In this Unit we will introduce the cross product along with a number of applications,
and we will compare the dot product and the cross product.
0. Determinants
For those students who have never dealt with matrices, we will recount the notion of a
deter
minant
.
Because we will be utilizing matrices only as a mnemonic device for the definition
of vector cross product, and not in their more basic role as representations of linear transfor
mations, we will direct our attention to the definition of the determinant of a 3
×
3 square
matrix.
A matrix (over the real numbers) is a rectangular array of numbers.
A square matrix is a
square array of numbers. For example,
A
=
2
1
3
5

12
0

1
4
7
is a 3
×
3 square matrix. Its
rows
are the horizontal entries, such as ( 2
1
3 ) and its
columns
are the vertical entries, such as
2
5

1
The determinant of a square matrix is a number. If
B
=
a
1
a
2
b
1
b
2
¶
is any 2
×
2 matrix,
its determinant is the number det
B
=
a
1
b
2

a
2
b
1
, that is to say, the product of the diagonal
elements minus the product of the antidiagonal elements. This defines the determinant only
for 2
×
2 matrices.
For a 3
×
3 matrix, the determinant of the matrix can be defined in terms of the determinants
of 2
×
2 matrices created from some of the elements of the 3
×
3 matrix in a fashion called
expanding along the first row
:
det
a
1
a
2
a
3
b
1
b
2
b
3
c
1
c
2
c
3
= +
a
1
det
b
2
b
3
c
2
c
3
¶

a
2
b
1
b
3
c
1
c
3
¶
+
a
3
b
1
b
2
c
1
c
2
¶
=
(1)
=
a
1
(
b
2
c
3

b
3
c
2
)

a
2
(
b
1
c
3

b
3
c
1
) +
a
3
(
b
1
c
2

b
2
c
1
)
(2)
where the definition of the determinant for 2
×
2 matrices has been used in the last step.
Take care to observe the minus sign before the entry
a
2
. In expanding along the first row, an
additional minus sign is introduced with the second entry of the first row. Note that the 2
×
2
submatrices
are obtained from the 3
×
3 matrix by eliminating the row and the column of each
element of the first row in turn. In linear algebra courses, it is taught that the determinant
may be obtained by expanding along any single row or any single column of a square matrix, in
a manner analogous to that above (although the additional minus signs may occur on different
elements). However, we will encounter only 3
×
3 matrices and will always expand along the
first row.
VECTOR GEOMETRY
74
GREENBERG
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For any square matrix
A
, the socalled inverse
A

1
exists if and only if det
A
6
= 0. A more
complete explanation of the meaning of the number det
A
is somewhat complicated. Suffice it
to say that for a 3
×
3 matrix
A
,

det
A

will be the volume of the parallelopiped whose edges
are the three vectors formed from the rows of
A
. In any case, for this course we shall only need
to know how to find the determinant.
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 Spring '08
 DONTREMEMBER
 Geometry, Determinant, Dot Product

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