Complex Numbers - Examples | ExamSolutions

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[@footballgames50]

Why cant we use (2√3 -2) as side b and 2 as side and then find the modulus or resultant AB, then multiply by 2 to get 2AB and then add the qnswer to OB?? PLS SOMEONE HELP, AM CONFUSED

[@humayunrashid8351]

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[@沈博智-x5y]

The way I did it was implementing to use of vectors. {i'm not suggesting this is a good method, because it's terrible} can represent the complex number u as a vector (1, 2sqrt(3)) represent complex number v = (3,2) i drew a sketch to see if I could find lengths AB and BC in terms of vectors. let vector "u" be denoted as vector OA and vector "v" denoted as vector OB notice vector OB - vector OA = vector AB thus (3,2) - (1,2sqrt(3) = (2, 2-2sqrt(3)) we see vector BC is perpendicular to AB thus the dot products of BC and AB must equal 0. let vector BC = (a*t,a*w) therefore (at,aw) dot (2,2-2sqrt(3)) = 2at + 2aw -2awsqrt(3) = 0 let t arbitarily be 1 therefore 2a(1) + 2aw - 2awsqrt(3) = 0 2aw(1-sqrt(3)) = -2a w = -1/(1-sqrt(3) = 1/(sqrt(3)-1) Thus, Vector BC = (a, a/(sqrt(3)-1)) Now we need to find the "a" value such that BC = 2AB length of vector BC = sqrt(a^2(1)^2 + a^2w^2) = sqrt(a^2(1+w^2))) = |a|*sqrt(1+(1/(sqrt(3)-1)^2) length of vector AB = sqrt(4+(2-2sqrt(3))^2) => length of vector 2AB = 2sqrt(4+(2-2sqrt(3))^2) Set BC = 2AB |a|*sqrt(1+(1/(sqrt(3)-1)^2) = 2sqrt(4+(2-2sqrt(3))^2) |a| = 2sqrt(4+(2-2sqrt(3))^2)/(sqrt(1+(1/(sqrt(3)-1)^2)) = 2sqrt(4+4-8sqrt(3)+12)/(sqrt(((sqrt(3)-1)^2 + 1)/((sqrt(3)-1)^2)) = 2sqrt(20-8sqrt(3))/(sqrt(3-2sqrt(3)+1+1)/(sqrt((sqrt(3)-1)^2)) = 2sqrt(4(5-2sqrt(3))/(sqrt(5-2sqrt(3))/(sqrt((sqrt(3)-1)^2)) = 2*2sqrt(5-2sqrt(3))/(sqrt(5-2sqrt(3))/(sqrt((sqrt(3)-1)^2)) |a| = 4*sqrt((sqrt(3)-1)^2) |a| = 4(sqrt(3) - 1) a = 4(sqrt(3)-1) or a = -4(sqrt(3)-1 Try a = -4(sqrt(3)-1 {spoiler alert, wrong choice} this implies vector BC is vector BC = (-4(sqrt(3)-1), -4) we are trying to find where complex number at point C is. We can represent this with vector OC, and Vector OC = OB + BC => Vector OC = (3,2) + (-4(sqrt(3)-1),-4) => Vector OC = (3-4(sqrt(3)-1), 2-4) => Vector OC = (3-12sqrt(3)+4,-2) => Vector OC = (7-12sqrt(3),-2) 7 - 12sqrt(3) < 0 and -2

[@rengokusannn8370]

wow, i tried the same thing but messed up in between, good job mann

[@terrencelevangama7430]

(2sqrt3 - 2 ) should be a imaginary part. please check

[@natanelkarp9826]

thats exactly what ive been thinking