Linear Algebra Solutions 53

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

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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 ) ...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online