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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.
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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.
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¢ JJ (^) 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 “−”.
We’ll illustrate this definition on the following pictures.
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a = (a 1 , a 2 )
b = (b 1 , b 2 )
From a to b we’re rotating in clockwise direction, so sign is “−”.
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¢∏ J]J
a = (a 1 , a 2 ) b = (b 1 , b 2 )
From a to b we’re rotating in counterclockwise direction, so sign is “+”.
6 A A A AU
≠¿≠
a = (a 1 , a 2 )
b = (b 1 , b 2 )
From a to b we’re rotating in clockwise direction, so sign is “−”.
Now we’ll state properties of the oriented area.
- area(a, b) = − area(b, a). This property follows from the fact that if we turn from a to b clockwise, then from b to a we turn counterclockwise and other way round, so signs are different, and absolute values will be the same since the parallelogram doesn’t change. From this property it follows that area(a, a) = 0 for any a:
area(a, a) = − area(a, a) ⇔ area(a, a) = 0.
area(e 1 , e 2 ) = 1. This is an area of the unit square, so it is equal to 1:
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x
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e (^26)
- (a) area(a 1 + a 2 , b) = area(a 1 , b) + area(a 2 , b). This property we can illustrate using the following picture.
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a 1 ∂
b
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a 2 §
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a 1 + a 2 ¢
h 1
h 2
So, area(a 1 , b) = −h 1 b, then area(a 2 , b) = −h 2 b, and area(a 1 + a 2 , b) = −(h 1 + h 2 )b, so this property holds.
Geometrically, determinant of a 3 × 3-matrix is the volume of the parallelepiped based on vectors (a 1 , a 2 , a 3 ), (b 1 , b 2 , b 3 ), and (c 1 , c 2 , c 3 ). We’ll give a mnemonic rule of writing determinants of a 3 × 3-matrix. We will rewrite first two columns of a matrix at the end of it, and then take following products with “+” signs:
c 1 c 2 c 3 c 1 c 2
b 1 b 2 b 3 b 1 b 2
a@ 1 a 2 a 3 a 1 a 2 @ @ @@
@ @ @ @@
@ @ @ @@
and following products with “−” signs:
c 1 c 2 c 3 c 1 c 2
b 1 b 2 b 3 b 1 b 2
a 1 a (^2) °a 3 a 1 a 2 ° ° °°
° ° ° °°
° ° ° °°
Example 1.5.
det
Geometrically this result means that vectors (1, 2 , 3), (4, 5 , 6), and (7, 8 , 9) are in the same plane, so the volume of a parallelepiped based on them is equal to 0.
