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Unformatted text preview: Lecture 19 Andrei Antonenko March 24, 2003 1 Area of the parallelogram Lets 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 lets 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 lets 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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This note was uploaded on 05/28/2011 for the course AMS 2010 taught by Professor Andant during the Spring '03 term at SUNY Stony Brook.
- Spring '03