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amanda522002's Question
Math
Posted 4 months ago

5. Points AA and BB are the representations in the complex plane of the numbers z=2−2iz=2-2 \mathrm{i} and w=−1−3iw=-1-\sqrt{3} \mathrm{i} respectively.
(a) Given that the origin is O\mathrm{O}, find the angle AOBA O B in radians, expressing your answer in terms of π\pi.
(b) Calculate the argument of zwz w in radians, again expressing your answer in terms of π\pi.
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Answer from Sia
Posted 4 months ago
Solution by Steps
step 1
Identify the complex numbers zz and ww in the complex plane. z=2−2iz = 2 - 2i and w=−1−3iw = -1 - \sqrt{3}i
step 2
Calculate the arguments of zz and ww. The argument of zz is arg⁡(z)=tan⁡−1(−22)=tan⁡−1(−1)=−π4\arg(z) = \tan^{-1}\left(\frac{-2}{2}\right) = \tan^{-1}(-1) = -\frac{\pi}{4}
step 3
The argument of ww is arg⁡(w)=tan⁡−1(−3−1)=tan⁡−1(3)=−π3\arg(w) = \tan^{-1}\left(\frac{-\sqrt{3}}{-1}\right) = \tan^{-1}(\sqrt{3}) = -\frac{\pi}{3}
step 4
Find the angle AOBAOB by taking the difference of the arguments: θ=arg⁡(w)−arg⁡(z)=−π3−(−π4)=−π3+π4=−4π12+3π12=−π12\theta = \arg(w) - \arg(z) = -\frac{\pi}{3} - (-\frac{\pi}{4}) = -\frac{\pi}{3} + \frac{\pi}{4} = -\frac{4\pi}{12} + \frac{3\pi}{12} = -\frac{\pi}{12}


 Part (b)
step 1
Calculate the product zwzw. zw=(2−2i)(−1−3i)zw = (2 - 2i)(-1 - \sqrt{3}i)
step 2
Expand the product: zw=2(−1)+2(−3i)−2i(−1)−2i(−3i)=−2−23i+2i+23=−2+23−23i+2izw = 2(-1) + 2(-\sqrt{3}i) - 2i(-1) - 2i(-\sqrt{3}i) = -2 - 2\sqrt{3}i + 2i + 2\sqrt{3} = -2 + 2\sqrt{3} - 2\sqrt{3}i + 2i
step 3
Simplify the expression: zw=−2+23−23i+2izw = -2 + 2\sqrt{3} - 2\sqrt{3}i + 2i
step 4
Calculate the argument of zwzw. The argument of zwzw is arg⁡(zw)=tan⁡−1(2i−23i−2+23)=tan⁡−1(2i(1−3)2(1−3))=tan⁡−1(i)=π2\arg(zw) = \tan^{-1}\left(\frac{2i - 2\sqrt{3}i}{-2 + 2\sqrt{3}}\right) = \tan^{-1}\left(\frac{2i(1 - \sqrt{3})}{2(1 - \sqrt{3})}\right) = \tan^{-1}(i) = \frac{\pi}{2}
Answer
(a) The angle AOBAOB is −π12-\frac{\pi}{12} radians.
(b) The argument of zwzw is π2\frac{\pi}{2} radians.
Key Concept
Complex numbers and their arguments in the complex plane
Explanation
The argument of a complex number is the angle it makes with the positive real axis, and the angle between two complex numbers can be found by taking the difference of their arguments.

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