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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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