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Unformatted text preview: Study sheet — Complex Numbers Harry Dankowicz Mechanical Science and Engineering University of Illinois at UrbanaChampaign [email protected] Objective: To investigate the relationship between transformations of the plane and complex numbers. Observation 1 Consider a transformation T that rotates objects in the plane by 30 ◦ and scales their size by a factor of 5 3 . Suppose, for example, that the vector v makes an angle 45 ◦ relative to the positive horizontal axis of some coordinate system and has length 2 . Then, the result of applying the transfor mation T to the vector v is the vector T ( v ) that makes an angle 45 ◦ + 30 ◦ = 75 ◦ relative to the same axis and has length 2 · 5 3 = 10 3 . The matrix of projections of the vector v onto the coordinate axis equals μ 2 cos 45 ◦ 2 sin 45 ◦ ¶ . (1) Similarly, the matrix of projections of the vector T ( v ) onto the coordinate axes equals μ 10 3 cos 75 ◦ 10 3 sin 75 ◦ ¶ . (2) But, μ 10 3 cos 75 ◦ 10 3 sin 75 ◦ ¶ = 10 3 μ cos (30 ◦ + 45 ◦ ) sin (30 ◦ + 45 ◦ ) ¶ = μ 5 3 · 2 ¶μ cos 30 ◦ cos 45 ◦ − sin 30 ◦ sin 45 ◦ sin 30 ◦ cos 45 ◦ + cos 30 ◦ sin 45 ◦ ¶ = μ 5 3 cos 30 ◦ − 5 3 sin 30 ◦ 5 3 sin 30 ◦ 5 3 cos 30 ◦ ¶μ 2 cos 45 ◦ 2 sin 45 ◦ ¶ , (3) i.e., the matrix of projections of T ( v ) onto the coordinate axes is obtained by premultiplying the matrix of projections of v onto the coordinate axes with the matrix μ 5 3 cos 30 ◦ − 5 3 sin 30 ◦ 5 3 sin 30 ◦ 5 3 cos 30 ◦ ¶ . (4) HHHHH Observation 2 Consider a transformation T that rotates objects in the plane by 30 ◦ and scales their size by a factor of 5 3 . Suppose, for example, that the vectors v and w make angles 30 ◦ and − 30 ◦ , respec tively, relative to the positive horizontal axis of some coordinate system and have lengths 1 2 and 3 2 , 1 respectively. Then, the results of applying the transformation T to the vectors v and w are the vectors T ( v ) and T ( w ) that make angles 60 ◦ and ◦ , respectively, relative to the same axis and have lengths 5 6 and 5 2 , respectively. The vector sum v + w makes an angle ≈ − 16 . 10 ◦ relative to the positive horizontal axis of the co ordinate system and has length √ 13 2 . The vector sum T ( v ) + T ( w ) makes an angle ≈ 13 . 90 ◦ = − 16 . 10 ◦ + 30 ◦ relative to the same axis and has length 5 √ 13 6 = √ 13 2 · 5 3 . It follows that, in this case, T ( v ) + T ( w ) = T ( v + w ) . (5) Theory: Denote by T ( r, θ ) a transformation that rotates objects in the plane by an angle θ and scales their size by a factor r . Then, T ( r, θ ) is a linear transformation , i.e., T ( r, θ ) ( α v + β w ) = α T ( r, θ ) ( v ) + β T ( r, θ ) ( w ) , (6) where v and w are arbitrary vectors and α and β are arbitrary real numbers....
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This note was uploaded on 11/17/2011 for the course TAM 412 taught by Professor Weaver during the Spring '08 term at University of Illinois, Urbana Champaign.
 Spring '08
 Weaver

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