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Unformatted text preview: CALCULUS II Spring 2009 LECTURE 7 This lecture provides • definition of a vector in R 2 • definition of a line in R 2 • parametric model of a line containing two given points • conversion of a parametric model to a scalar model of a line • definition of a vectorvalued function • definition of a parametric curve • parametric model of a circle • two different parametric models of the same pointset • parametric model of y = x 1 3 ,∞ < x < ∞ . 1 Recall We recall from Lecture 6. • a model for areas of surfaces of revolution • several examples of surface area computation End of Recall 2 Vectors Vectors in the Plane . We recall that elements of R 2 (of the Euclidean plane) are called vectors . For example the elements (2 , 1) and ( √ 3 , 21) are vectors. Vectors can be added: (2 , 1) + ( √ 3 , 21) = (2 + √ 3 , 20) . This is called a vector sum . Vectors can be multiplied by a scalar (by a real number): √ 3( √ 3 , 21) = (3 , 21 √ 3) . This product is called the scalar product . It is common to denote a vector in the plane by a single symbol. For example if a = (2 , 1) and b = ( √ 3 , 21), then a + b = (2 + √ 3 , 20) and √ 3 b = (3 , 21 √ 3) . Vectors have lengths. If a = (2 , 1) and b = ( √ 3 , 21) then the lengths of a and b are denoted by  a  and  b  respectively. They are given by  a  = p 2 2 + ( 1) 2 and  b  = q ( √ 3) 2 + (21) 2 . That is  a  = √ 5 and  b  = 12 . The length of a vector is sometimes called its magnitude . 3 If u = (2 , 3) and v = ( 1 , 2), the vector difference between u and v is a vector; it is given by u v = (2 , 3) ( 1 , 2) = (2 + 1 , 3 + 2) = (3 , 5) . Thus u v = (3 , 5). (We have interpreted the expression ( 1 , 2) to mean +( 1)( 1 , 2) = +(1 , 2).) The magnitude of the difference between u and v is written  u v  and is given by  u v  = p 3 2 + 5 2 = √ 34 . One notes that we have used the expression  a  to denote the length of a vec tor a . We use the same expression  a  to denote the absolute value of the real number a . It is almost always clear from the context whether or not the ver tical bars represent length of a vector or the absolute value of a real number. We have not mentioned the notion of multiplication of vectors. There is a commonly used notion of the product of two vectors, called the dot product . Here the product is not another vector, but instead a real number. More about that later. During our investigation of vectors in R 3 we introduce the notion of cross product . However, for the time being, we limit our notions of product to the scalar product, defined above. We have declared that elements of R 2 are vectors. So a single point plotted on the x,yplane is a model of a vector....
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This note was uploaded on 02/09/2009 for the course MATH 1020 taught by Professor Ecker during the Spring '08 term at Rensselaer Polytechnic Institute.
 Spring '08
 ECKER
 Calculus, Scalar

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