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Question
Find the modulus and amplitude of the following complex numbers.
−3(1 − i)
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Solution
Let z = −3(1 − i) = −3 + 3i
∴ a = −3, b = 3, i.e. a < 0, b > 0
∴ |z| = `sqrt("a"^2 + "b"^2)`
= `sqrt((-3)^2 + 3^2)`
= `sqrt(9 + 9)`
= `sqrt(18)`
= `3sqrt(2)`
Here, (−3, 3) lies in 2nd quadrant.
∴ amp (z) = `tan^-1("b"/"a") + pi`
= `tan^-1(3/(-3)) + pi`
= tan−1(−1) + π
= − tan−1(1) + π
= `-pi/4 + pi`
= `(-pi)/4 + pi`
= `(3pi)/4`
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