Lecture 19
Andrei Antonenko
March 24, 2003
1
Area of the parallelogram
Let’s consider a plane
R
2
.
Now we will consider parallelograms on this plane, and compute
their area.
First thing which is clear from elementary geometry is a formula for the area of the paral
lelogram.
•
•
•
•
•
•
•
•
•
•
a
h
The area of the parallelogram is equal to the product of the base band the height,
S
=
ah
.
Now let’s consider a plane
R
2
as a vector space, and let we have 2 vectors
a
= (
a
1
, a
2
) and
b
= (
b
1
, b
2
) on the plane.

6
¢
¢
¢
¢
‚
J
J]
¢
¢
¢
¢
J
J
a
= (
a
1
, a
2
)
b
= (
b
1
, b
2
)
Now with this pair of vectors we can associate a parallelogram, as shown on the picture above.
Out main goal is to study the properties of the area of this parallelogram and compute it in
terms of vectors
a
= (
a
1
, a
2
) and
b
= (
b
1
, b
2
).
First let’s give a definition of the oriented area of the parallelogram.
Definition 1.1.
The
oriented area
area(
a, b
)
of the parallelogram based on two vectors
a
= (
a
1
, a
2
)
and
b
= (
b
1
, b
2
)
is the standard geometrical
area of it taken with appropriate sign.
The sign is determined by the following rule.
If the
rotation from
a
to
b
(by the smaller angle) goes counterclockwise, then the sign is “
+
”, otherwise,
the sign is “

”.
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 Spring '03
 Andant
 Pallavolo Modena, Sisley Volley Treviso, Associazione Sportiva Volley Lube, Clockwise, Piemonte Volley

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