Question :
Write the following in the rectangular form:
(i) [(5 + 9i) + (2 − 4i)] whole bar
Solution :
[(5 + 9i) + (2 − 4i)] whole bar = (5 + 9i) bar + (2 − 4i) bar
= (5 - 9i) + (2 + 4i)
= (5 + 2) + (-9i + 4i)
= 7 - 5i
(ii) (10 - 5i)/(6 + 2i)
Solution :
(10 - 5i)/(6 + 2i)
Multiplying both numerator and denominator by the conjugate of of denominator, we get
= [(10 - 5i)/(6 + 2i)] [(6 - 2i)/(6 - 2i)]
= (10 - 5i) (6 - 2i) / (36 - 4(-1))
= (60 - 20i - 30i + 10 (-1)) / (36 + 4)
= (60 - 20i - 30i - 10) / 40
= (50 - 50i) / 40
= (5 - 5i)/4
= (5/4) (1 - i)
(iii) 3i bar + 1/(2 - i)
Solution :
3i bar + 1/(2 - i)
= - 3i + { (1/(2 - i)) ((2 + i)/(2 + i)) }
= - 3i + { (2 + i)/(4 - (-1)) }
= - 3i + { (2 + i)/5 }
= (2/5) + {(1/5) - 3}
= (2/5) - (14/5)
Question 2 :
If z = x + iy , find the following in rectangular form.
(i) Re (1/z)
Solution :
z = x + iy
1/z = 1/(x + iy)
= [1/(x + iy)] [(x - iy)/(x - iy)]
= (x - iy)/(x2 + y2)
= (x/(x2 + y2)) - i (y/(x2 + y2))
Hence the Re (1/z) is (x/(x2 + y2)) - i (y/(x2 + y2)).
(ii) Re (i z bar)
Solution :
= i (x + iy) bar
= i(x - iy)
= ix - i2y
= ix + y
Hence the Re (i z bar) is y.
(iii) Im(3z + 4zbar − 4i)
Solution :
(3z + 4zbar − 4i) = [3(x + iy) + 4(x + iy) bar - 4i]
= 3x + i3y + 4(x - iy) - 4i]
= 3x + i3y + 4(x - iy) - 4i
= 3x + i3y + 4x - i4y - 4i
= (3x + 4x) + i(-4y + 3y - 4)
= (3x + 4x) + i(-y - 4)
Hence the value of Im(3z + 4zbar − 4i) is - y - 4.
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