Linear Algebra Solutions 53

Linear Algebra Solutions 53 - y (ii) The number lies in the...

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Unformatted text preview: y (ii) The number lies in the third quadrant of the complex plane. −3 − i |−3 − i| = 2 2 = 1 2 √ 10 1√ 2 + (−1)2 = (−3) 9+1= . 2 2 T ' −3−i 2 c Also Arg ( −3−i ) = −π + α, where tan α = 2 13 / 2 = 1/3 and 0 < α < π/2. Hence α = 2 tan −1 (1/3). (iii) The number lies in the second quadrant of the complex plane. √ | − 1 + 2i| = (−1)2 + 22 = 5. Ex α A  −1 + 2i y u e e e T e α ee ' Ex Also Arg (−1 + 2i) = π − α, where tan α = 2 and 0 < α < π/2. Hence α = tan −1 2. −1 2 (iv) The number lies in the second quadrant of the complex plane. = √ √ | − 1 + i 3| −1 + i 3 = 2 2 √ 1 1√ (−1)2 + ( 3)2 = 1 + 3 = 1. 2 2 Also Arg ( −1 + 2 √ tan α = 23 / 1 = 2 Hence α = π/3. √ 3 2 i) √ + “ t √ 3 2i t t ' c y T t αtt Ex c = π − α, where 3 and 0 < α < π/2. √ √ 3i)( 3 − i). Then √√ |z | = |1 + i||1 + 3i|| 3 − i| √ √ = 12 + 12 12 + ( 3)2 ( 3)2 + (−1)2 √√√ √ = 2 4 4 = 4 2. 6. (i) Let z = (1 + i)(1 + Arg z ≡ Arg (1 + i) + Arg (1 + 56 √ √ 3) + Arg ( 3 − i) (mod 2π ) ...
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