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03-24-03

# 03-24-03 - Lecture 19 Andrei Antonenko 1 Area of the...

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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. 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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03-24-03 - Lecture 19 Andrei Antonenko 1 Area of the...

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