Question #347280

2. Given z1 = 2∠45o


, ; z2 = 3∠120o and z3 = 4∠180o


. Determine the following and leave your


answers in rectangular form:


(i)


(z1)2 +z2


z2 +z3


(5)


(ii)


z1


z2z3


1
Expert's answer
2022-06-03T11:25:02-0400

(i)

(z1)2=(2)2(245°)=4i(z_1)^2=(2)^2\angle(2\cdot45\degree)=4i

z2+z3=3(12+32i)4=112332i)z_2+z_3=3(-\dfrac{1}{2}+\dfrac{\sqrt{3}}{2}i)-4=-\dfrac{11}{2}-\dfrac{3\sqrt{3}}{2}i)

z2z2+z3=3+33i1133i\dfrac{z_2}{z_2+z_3}=\dfrac{-3+3\sqrt{3}i}{-11-3\sqrt{3}i}

=(3+33i)(1133i)121+27=\dfrac{(-3+3\sqrt{3}i)(-11-3\sqrt{3}i)}{121+27}


=33+93i333i+27148=\dfrac{33+9\sqrt{3}i-33\sqrt{3}i+27}{148}


=15376337i=\dfrac{15}{37}-\dfrac{6\sqrt{3}}{37}i


(z1)2+z2z2+z3=4i+15376337i(z_1)^2+\dfrac{z_2}{z_2+z_3}=4i+\dfrac{15}{37}-\dfrac{6\sqrt{3}}{37}i

=1537+1486337i=\dfrac{15}{37}+\dfrac{148-6\sqrt{3}}{37}i

(ii)


z2z3=3(4)(120°+180°)=663iz_2z_3=3(4)\angle(120\degree+180\degree)=6-6\sqrt{3}i

z1z2z3=2+2i663i\dfrac{z_1}{z_2z_3}=\dfrac{\sqrt{2}+\sqrt{2}i}{6-6\sqrt{3}i}

=(2+2i)(6+63i)36+108=\dfrac{(\sqrt{2}+\sqrt{2}i)(6+6\sqrt{3}i)}{36+108}

=62+66i+62i66144=\dfrac{6\sqrt{2}+6\sqrt{6}i+6\sqrt{2}i-6\sqrt{6}}{144}

=6224+6+224i=-\dfrac{\sqrt{6}-\sqrt{2}}{24}+\dfrac{\sqrt{6}+\sqrt{2}}{24}i


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