EAS 208 (Fall 2010)
Cross Product
Definition:
The cross product is an operation between two vectors in threedimensional Euclidean
space. The cross product of vector
and vector
, denoted as
u
v
×
uv
, is defined as:
ˆ
if
,
0,
0, and
sin
if
0, or
0
θ
θπ
≠≠
≠
⎧⋅⋅
×=
⎨
=
==
=
⎩
e
where
is the interior angle between
u
and
, and
is a unit vector normal to the
plane of
u
and
, directed such that the vectors
,
,
form a righthanded system.
Note, that the result of the cross product
v
ˆ
e
v
u
v
ˆ
e
×
is a vector.
Figure 1
Cross Product of vector
u
and vector
v
Properties:
( )
− ×
vu
(anticommutativity)
()
( )
aa
×
uv uv
(associativity)
( ) ( )
+×=× +×
uv w uw vw
(distributivity)
1
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View Full DocumentGeometric Significance:
The magnitude of the cross product
×
uv
is equal to the area of the
u
,
parallelogram
as illustrated in Figure (2) (
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 Fall '08
 MehdiAhmadizadeh
 Dot Product, Euclidean geometry

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