# Complex Numbers finding real

## Homework Statement

u = -1 + j$\sqrt{3}$
v = $\sqrt{3}$ - j

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

|u-$a/v$| = 2$\sqrt{2}$

## 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$\sqrt{3}$ + 12j. However, this does not work and is not a real scalar either...

This problem should be able to be done by hand.

NascentOxygen
Staff Emeritus
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?

u = -1 + j$\sqrt{3}$
v = $\sqrt{3}$ - j
|u - a/v| = 2$\sqrt{2}$

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

$\sqrt{u - a/v}$ = 2$\sqrt{2}$

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

Substitute in v and u and begin performing basic algebra.

($\sqrt{3}$ - j) * (-1 + j$\sqrt{3}$ - 8) = a
($\sqrt{3}$ - j) * (-9 + j$\sqrt{3}$) = a

Then FOIL the binomial
-9$\sqrt{3}$ + 3j + 9j + $\sqrt{3}$
a = -8$\sqrt{3}$ + 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$\sqrt{2}$ can easily be expressed using trigonometric functions (sin($\frac{\pi}{4}$) and cos($\frac{\pi}{4}$))but I don't know how this can play a part.

Last edited:
gneill
Mentor
u = -1 + j$\sqrt{3}$
v = $\sqrt{3}$ - j

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

$\sqrt{u - a/v}$ = 2$\sqrt{2}$
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...

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.