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Unformatted text preview: a ∈ R n has length one, that is  a  = 1 . Deﬁnition 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 coeﬃcients 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 a2 b . Remark 1.3. One can also use vectors to describe lines and planes. As an example, the xy plane is the set of all points α i + β j where α , β ∈ R . Simliarly the yz plane is the set of all points βj + γ k where β , γ ∈ R . 2...
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 Winter '08
 JackWaddell
 Linear Algebra, Vector Calculus, Vectors, Standard basis, Tromba Vector Calculus

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