Author Topic: Complex Numbers Question (Read 1013 times) Tweet Share

ice_blockie · « **on:** February 19, 2008, 09:56:30 pm »

On an Argand diagram, $O$ is the origin and $P$ represents the point $3 + i$ . The point $Q$ represents $a+bi$ , where $a$ and $b$ are both positive. If triangle $OPQ$ is equilateral, find $a$ and $b$ .

I think you could find the polar form of $3+i$ and then add 60 degrees to the angle to get the coordinates for $Q$ . But it doesnt seem to work. Help?

Collin Li · « **Reply #1 on:** February 19, 2008, 10:08:07 pm »

To find $Q$ , we must multiply the complex number $3+i$ by $\mbox{cis} 60\textdegree$ or $\mbox{cis} -60\textdegree$

$\mbox{cis} 60\textdegree = \frac{1}{2} + i\frac{\sqrt{3}}{2} = \frac{1}{2}\left(1 + \sqrt{3}i\right)$

Therefore, $q = \frac{1}{2}(3+i)\left(1 + \sqrt{3}i\right) = \frac{1}{2}\left[3 - \sqrt{3} + i\left(1 + 3\sqrt{3}\right)\right]$

$a = \frac{3}{2} - \frac{\sqrt{3}}{2}$

$b = \frac{1}{2} + \frac{3\sqrt{3}}{2}$

Multiplying by $\mbox{cis} -60\textdegree$ would result in negative values of $a$ and $b$

gta007 · « **Reply #2 on:** February 25, 2008, 10:55:01 pm »

Coblin, you forgot to multiply by 1/2 in the end. lol

Collin Li · « **Reply #3 on:** February 25, 2008, 11:04:06 pm »

LOL you're right. It is fixed now.

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Author Topic: Complex Numbers Question (Read 1013 times) Tweet Share

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