Lecture_7 - CALCULUS II Spring 2009 LECTURE 7 This lecture provides • definition of a vector in R 2 • definition of a line in R 2 •

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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 vector-valued function • definition of a parametric curve • parametric model of a circle • two different parametric models of the same point-set • 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,y-plane 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.

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Lecture_7 - CALCULUS II Spring 2009 LECTURE 7 This lecture provides • definition of a vector in R 2 • definition of a line in R 2 •

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