12.4 The Cross Product
The cross product is a
vector,
sometimes called the
vector product.
Cross products are only
found with 3-dimensional vectors.
Cross Product
1,
2
3
1
2
3
2
3
3
2
3
1
1
3
1
2
2
1
,
,
,
,
,
a a
a
b b
b
a b
a b
a b
a b
a b
a b
=
=
×
=
−
−
−
a
b
a
b
Determinants can be used to find this product. The TI-85 and TI-86 will also perform this
operation.
Determinants and Cross Product
1
2
3
1
2
3
2
3
1
3
1
2
2
3
1
3
1
2
1
2
3
1
2
3
,
,
,
,
a
b
ad
bc
c
d
a
a
a
b b
b
a
a
a
a
a
a
b
b
b
b
b
b
a
a
a
b
b
b
=
−
=
=
×
=
−
+
×
=
a
b
a
b
i
j
k
i
j
k
a
b
Example: Find the cross product for
1,2,
3
5,
1,
2
=
−
=
−
−
a
b
.
Theorems, etc.
(a)
The cross product is orthogonal to both
a
and
b.
(b)
If
θ
is the angle between
a
and
b
then
sin
θ
×
=
a
b
a
b
.
(c)
Two nonzero vectors
a
and
b
are parallel if and only if the cross product is 0.

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