3
5
Rev.F09
The Cross Product of Two Vectors in
ℜ
3
http://faculty.valenciacc.edu/ashaw/
Click link to download other modules.
•
The
cross product
of two vectors
u
= (
u
1
,u
2
,u
3
) and
v
=
(
v
1
,v
2
,v
3
),
u
x
v,
in
ℜ
3
is a vector in
ℜ
3
.
•
The direction of the
cross product
,
u
x
v
, is always
perpendicular to the two vectors
u
and
v
and the plane
determined by
u
and
v
that is parallel to both
u
and
v
.
•
The norm of the cross product is
u
x
v
u
v
!
u
!
!
v
=
!
u
!
v
sin(
"
).
6
Rev.F09
The Cross Product of Two Vectors in
ℜ
3
(Cont.)
http://faculty.valenciacc.edu/ashaw/
Click link to download other modules.
The
cross product
can be represented symbolically in
the form of a 3 x 3 determinant:
u
x
v
=
where
i
= (1,0,0),
j
= (0,1,0),
k
= (0,0,1) are
standard
unit vectors
.
i
j
k
u
1
u
2
u
3
v
1
v
2
v
3
=
u
2
u
3
v
2
v
3
i
!
u
1
u
3
v
1
v
3
j
+
u
1
u
2
v
1
v
2
k
=
u
2
u
3
v
2
v
3
,
!
u
1
u
3
v
1
v
3
,
u
1
u
2
v
1
v
2
"
#
$
$
%
&
'
'
Note:
Every vector in
ℜ
3
is expressible in terms of the standard unit vectors.
v
= (
v
1
,v
2
,v
3
) =
v
1
(1,0,0) +
v
2
(0,1,0) +
v
3
(0,0,1) =
v
1
i
+
v
2
j
+
v
3
k