35 - a R n has length one, that is | a | = 1 . Denition...

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Lecture 6 January 17, 2012 Other References: Marsden and Tromba Vector Calculus Edwards, Advanced Calculus of Several Variables 1 Vectors Definition 1.1. A vector in R n , a is an ordered set of numbers ( a 1 ,...,a n ) where a i R 1 and α i are the components of a . Definition 1.2. If a and b are vectors in R n then the vector sum a + b = ( a 1 + b 1 ,a 2 + b 2 ,...,a n + b n ) Definition 1.3. If a is a vector in R n and α is a scalar, that is α R 1 , then the s calar multiple or scalar product is defined by α a = ( αa 1 ,αa 2 ,...,αa n ) . Corollary 1.1. If a and b are vectors in R n and α and β are scalars then i) ( αβ ) a = α ( β a ) – associativity ii) ( α + β ) a = α a + β a – distributivity iii) α ( a + b ) = α a + α b – distributivity iv) a + 0 = a , with 0 being unique v) 0 a = 0 for all a – property of the zero element vi) 1 a = a for all a – property of the identity For a deeper look at this see Foundations of Analysis by Landau. 1
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Remark 1.1. Sets with the above properties are called vector spaces and there are more general vector spaces than R n . Definition 1.4. The length of a vector a R n is defined as | a | = v u u t n X i =1 a 2 i . Definition 1.5. A unit vector
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Unformatted text preview: a R n has length one, that is | a | = 1 . Denition 1.6. The standard basis in R 3 is denoted by i , j , k is given by i = (1 , , 0) , j = (0 , 1 , 0) , k = (0 , , 1) ADD PICTURE Remark 1.2. i , j and k are unit vectors. Example 1.1. i + j = j + i . ADD PICTURE Lemma 1.1. Any vector in R 3 can be decomposed into a sum of scalar coecients multiplying the three standard basis vectors. Proof. Algebraic: Let a = ( a 1 ,a 2 ,a 3 ) be a vector in R n . Then a = a 1 e 1 + a 2 e 2 + a 3 e 3 . Geometric:: ADD PICTURE Example 1.2. If a = 2 i + 3 j and b =-j + 2 k sketch 2 a-2 b . Remark 1.3. One can also use vectors to describe lines and planes. As an exam-ple, the xy plane is the set of all points i + j where , R . Simliarly the yz plane is the set of all points j + k where , R . 2...
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35 - a R n has length one, that is | a | = 1 . Denition...

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