Loser66
  • Loser66
How to prove perpendicular lines in complex plane? Question: Let w = a+ bi not 0. Use Euclidean geometry to explain why the set {z in C : |z-w| =|z+w|} is a straight line through the origin 0 perpendicular to the line through -w and w. Then verify the result analytically. I am done with part a and most of part b until Re (z w(bar)) =0, then stuck Please, help
Mathematics
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schrodinger
  • schrodinger
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Loser66
  • Loser66
Here is my work: |z -w| = |z +w| \(\implies |z-w|^2 =|z+w|^2\) Expand and simplify, I got \(4Re z\bar w=0\implies Re z\bar w=0\), then???
ganeshie8
  • ganeshie8
Let \(z=x+iy\) the required line is \(ax+by=0\)
ganeshie8
  • ganeshie8
the line through \(-w\) and \(w\) has a direction vector \((a, b)\)

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ganeshie8
  • ganeshie8
the line \(ax+by=0\) has a direction vector \((-b,a)\)
ganeshie8
  • ganeshie8
the dot product of direction vectors is \(0\), therefore the lines are perpendicular
Loser66
  • Loser66
I am sorry. I didn't see your reply. Thanks for the explanation. Question: how can we prove it passes through origin?
ganeshie8
  • ganeshie8
\(ax+by=0\) satisfies the point \((0,0)\)

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