Minimum of Complex sum of 3 terms

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\hspace{-16} $ Minimum value of $\bf{\left|z-1-i \right| + \left |z+2-3i \right| + \left |z+3+2i \right|}$\\\\\\ where $\bf{z = x+iy}$ and $\bf{i = \sqrt{-1}}$

#Mathematics #Algebra #Complex Numbers

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1 Answers

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Shaswata Roy ·2013-08-04 21:27:43

Plot the points A(1,1),B(-2,3) and C(-3,-2).

We want to find P(x,y) such that PA+PB+PC is minimum.

This point P is the Fermat's point.
The unique property of the Fermat point is that,
<APB=<APC=<BPC=120Â°

PA = e^{i 120Â°}PB

(1+i-x-iy)=(-12+√3i2)(-2+3i-x-iy)..(i)

and,

PB = e^{i 120Â°}PC

(-2+3i-x-iy)=(-12+√3i2)(-3-2i-x-iy)...(ii)

Solving i and ii you'll the required point.

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Anik Chatterjee i think it will be PB = ei 120Â°PA &PC = ei 120Â°PB..
Upvote·0· Reply ·2013-08-05 06:09:34
Shaswata Roy My bad!!The angles will be -120 and not 120.Or we can Interchange between PA,PB and PC like you've done.
Upvote·0· Reply ·2013-08-06 22:51:28
man111 singh Thanks Shaswata
Upvote·0· Reply ·2013-10-06 21:30:30
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