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complex_numbers.20100115.4b5099ac2012a9.06205154

complex_numbers.20100115.4b5099ac2012a9.06205154 - Study...

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Study sheet — Complex Numbers Harry Dankowicz Mechanical Science and Engineering University of Illinois at Urbana-Champaign [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
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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 0 , 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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