T
he
C
ross
P
roduct
When discussing the dot product, we showed how two vectors can be combined to get
a number. Here, we shall see another way of combining vectors, this time resulting in a
vector. This operation is called the
cross product
. The cross product of two vectors
v
and
w
is a third vector which is perpendicular to
both
v
and
w
.
Unlike the dot product (which is defined for any dimension of the vector), the cross
product is only defined for three-dimensional vectors. The cross product is more easily
remembered in terms of a linear algebra result, as the determinant of a 3
μ
3 matrix.
Cross Product
Let
1
2
3
v
v
v
v
i
j
k
and
1
2
3
w
w
w
w
i
j
k
. The cross product of
v
and
w
, denoted by
v
μ
w
, is a
vector
given by
1
2
3
2
3
3
2
3
1
1
3
1
2
2
1
1
2
3
(
)
(
)
(
v
v
v
v w
v w
v w
v w
v w
v w
w
w
w
i
j
k
v
w
i
)
j
k
Example 1:
Compute
v
μ
w
if
4
3
0
v
i
j
k
and
2
2
5
w
i
j
k
. Verify that
v
μ
w
is
perpendicular to both
v
and
w
.
Solution:
v
μ
w
= [(–3)(5) – (0)(2)]
i
+ [(0)(–2) – (4)(5)]
j
+ [(4)(2) – (–3)(–2)]
k
= –15
i
– 20
j
+ 2
k
.
Notice that
and
.
(4)( 15)
( 3)( 20)
(0)(2)
60
60
0
0
v
v
w
2)( 15)
(2)( 20)
(5)(2)
30
40
10
0
(
w
v
w
And so, we see that both
v
and
w
are perpendicular to
v
μ
w
.
Similar to the case of the dot product, there is a nice geometrical interpretation of the
cross product, this time in terms of areas of parallelograms. To that end, we make the
following claim.
1

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