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Complex Numbers finding real

  • Thread starter DmytriE
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  • #1
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Homework Statement



u = -1 + j[itex]\sqrt{3}[/itex]
v = [itex]\sqrt{3}[/itex] - j

Let a be a real scaling factor. Determine the value(s) of a such that

|u-[itex]a/v[/itex]| = 2[itex]\sqrt{2}[/itex]

Homework Equations



The equation above is the only relevant equation.

The Attempt at a Solution



I have converted the cartesian equation into polar in the hopes that it would be made easier but apparently not. I have gotten the following answer -8[itex]\sqrt{3}[/itex] + 12j. However, this does not work and is not a real scalar either...

This problem should be able to be done by hand.
 

Answers and Replies

  • #2
NascentOxygen
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You haven't shown any working, so it's difficult to point out where you went wrong.

Can you evaluate u-a/v and express it as a complex number?
 
  • #3
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u = -1 + j[itex]\sqrt{3}[/itex]
v = [itex]\sqrt{3}[/itex] - j
|u - a/v| = 2[itex]\sqrt{2}[/itex]

Here is what I have done step by step to get my answer.

[itex]\sqrt{u - a/v}[/itex] = 2[itex]\sqrt{2}[/itex]

u - a/v = 8
v(u - 8) = a

Substitute in v and u and begin performing basic algebra.

([itex]\sqrt{3}[/itex] - j) * (-1 + j[itex]\sqrt{3}[/itex] - 8) = a
([itex]\sqrt{3}[/itex] - j) * (-9 + j[itex]\sqrt{3}[/itex]) = a

Then FOIL the binomial
-9[itex]\sqrt{3}[/itex] + 3j + 9j + [itex]\sqrt{3}[/itex]
a = -8[itex]\sqrt{3}[/itex] + 12j

Now, this is the complex number that I get but this is not a real scalar. How should I proceed or should I begin trying something else? This is part of the section where a calculator is not needed.

Food for thought:
2[itex]\sqrt{2}[/itex] can easily be expressed using trigonometric functions (sin([itex]\frac{\pi}{4}[/itex]) and cos([itex]\frac{\pi}{4}[/itex]))but I don't know how this can play a part.
 
Last edited:
  • #4
gneill
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u = -1 + j[itex]\sqrt{3}[/itex]
v = [itex]\sqrt{3}[/itex] - j

Here is what I have done step by step to get my answer.

[itex]\sqrt{u - a/v}[/itex] = 2[itex]\sqrt{2}[/itex]
Ah, but that's not the magnitude of the expression. For a complex number z = x + y*j, the magnitude is given by

$$|z| = \sqrt{x^2 + y^2} $$

Start by expanding the expression u - a/v and collect into its real and imaginary parts (assume that a is a real number). Then apply the definition of the magnitude to the result. Note that you can clear the square root by taking the square on both sides...
 
  • #5
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Start by expanding the expression u - a/v and collect into its real and imaginary parts (assume that a is a real number). Then apply the definition of the magnitude to the result.
Yes! Thank you gneill! I was blinded by my continuous mistakes. The help that made it all clear.
 

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