82
CHAPTER 1. COORDINATES AND VECTORS
−→
σ
(
ABC
) =
σ
(
ABC
)
−→
k
: the oriented area is the vector
−→
k
times the signed
area in our old sense. This interpretation can be applied as well to any
oriented polygon contained in a plane in space.
In particular, by analogy with Proposition
1.6.1
, we can defne a Function
which assigns to a pair oF vectors
−→
v ,
−→
w
∈
R
3
a new vector representing the
oriented area oF the parallelogram with two oF its edges emanating From
the origin along
−→
v
and
−→
w
, and oriented in the direction oF the frst vector.
This is called the
cross product
16
oF
−→
v
and
−→
w
, and is denoted
−→
v
×
−→
w.
±or example, the sides emanating From
A
in
△
ABC
in ±igure
1.41
are
represented by
−→
v
=
−−→
AB
= 2
−→
ı
+
−→
+
−→
k
−→
w
=
−→
AC
=
−→
ı
+ 2
−→
−
−→
k
;
these vectors, along with the direction oF
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 Fall '08
 ALL
 Calculus, Vectors, Dot Product

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