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Unformatted text preview: Theorem 3.2.1: If v is a vector in R n , and if k is any scalar, then (a)  v  (b)  v  = 0 if and only if v=0 (c)  kv  =  k  v  Proof: Exercise, (Hint: express u as u = ( u 1 ,u 2 ,...,u n ) and use the definition of norm) Applying the definition of length to a vector between two points, one can find the distance between two points. Let P 1 = ( x 1 ,y 1 ) ,P 2 = ( x 2 ,y 2 ) then d ( P 1 ,P 2 ) =  ~ P 1 P 2  = p ( x 2 x 1 ) 2 + ( y 2 y 1 ) 2 . Definition: If P = ( u 1 ,u 2 ,...,u n ) and Q = ( v 1 ,v 2 ,...,v n ) are two points in R n then d ( P,Q ) =  ~ PQ  = p ( v 1 u 1 ) 2 + ... + ( v n u n ) 2 Unit vector: unit vector= vector with length 1 A way to obtain a unit vector: Let v be a vector, then v  v  will be a unit vector. Proof: Consider  v  v   =  1  v  v  . Note that 1  v  is a scalar and by Theorem 3.2.1 (c)  1  v  v  =  1  v   v  = 1  v   v  by Theorem 3.2.1 (a). Hence  v  v   = 1 in R n we have standard unit vectors e 1 = (1 , , ,..., 0) ,e 2 = (0 , 1 , , ,..., 0) ,e 3 = (0 , , 1 , ,..., 0) ,...,e n = (0 , ,..., , 1) . If v = ( v 1 ,v 2 ,...,v n ) is a vector in R n then v = v 1 e 1 + v 2 e 2 + ... + v n e n . That is any vector in R n can be written as a linear combination of standard unit vectors....
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This note was uploaded on 04/27/2011 for the course ENGR 130 taught by Professor Zhang during the Spring '11 term at University of Alberta.
 Spring '11
 Zhang

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