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JUMBLING TSPs
How is
ab c
¥
()
∑
related to
ac b
¥
∑
?
Determinant approach:
¥
∑=
aaa
bbb
ccc
123
¥
The determinant forms differ only by a single switch of two rows, so
they differ only by a sign.
¥

¥
∑
[]
Geometric approach:
Both
¥
∑
and
¥
∑
represent the volume of the
parallelepiped determined by the position vectors for
a
,
b
, and
c
. This
is consistent with the box above.
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View Full DocumentHow is
ab c
¥
()
∑
related to
bc a
¥
∑
?
Determinant approach:
¥
∑=
aaa
bbb
ccc
123
¥
The determinant forms differ by two switches of pairs of rows, so
there is a “double negative” effect, and the determinants are equal.
ab c bc a
¥
¥
∑
Geometric approach:
(Similar to the first example)
Remember that dot products are commutative, and cross products are
anticommutative, so it may be easy to relate some jumbles. For example,
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 Spring '11
 staff
 Math, Calculus, Determinant

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