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which is also an elegant relationship.
Furthermore, if two columns or rows of a determinant are interchanged, its sign is reversed.
Also, the value of a determinant is unaffected if its elements are transposed. Therefore,
x
a
y
a
z
a
x
a
y
a
z
a
x
b
y
b
z
b
x
b
y
b
z
b
=−
x
c
y
c
z
c
=−
x
a
y
a
z
a
x
c
y
c
z
c
x
b
y
b
z
b
x
c
y
c
z
c
and
acb
=−
abc
=−
a
·
c
×
b
·=−
b
·
a
×
c
Just to prove that Eq. (2.18) works, let's take the first example above, where
a
=−
k
b
=−
i
c
=
j
Using Eq. (2.18), we find
00
−
1
abc
=
a
·
b
×
c
=
−
100
010
=
1
We are now in a position to prove that
a
·
b
×
c
=
b
·
c
×
a
=
c
·
a
×
b
, even though we
concluded it in Eq. (2.14).
From Eq. (2.18), we have
x
a
y
a
z
a
a
·
b
×
c
=
x
b
y
b
z
b
x
c
y
c
z
c
It can be shown that exchanging two rows of a determinant changes its sign. Therefore,
x
a
y
a
z
a
x
b
y
b
z
b
x
b
y
b
z
b
x
b
y
b
z
b
=−
x
a
y
a
z
a
=
x
c
y
c
z
c
=
b
·
c
×
a
x
c
y
c
z
c
x
c
y
c
z
c
x
a
y
a
z
a
Similarly,
x
a
y
a
z
a
x
c
y
c
z
c
x
c
y
c
z
c
x
b
y
b
z
b
=−
x
b
y
b
z
b
=
x
a
y
a
z
a
=
·
×
c
a
b
x
c
y
c
z
c
x
a
y
a
z
a
x
b
y
b
z
b
Therefore,
a
·
b
×
c
=
b
·
c
×
a
=
c
·
a
×
b
(2.19)
