# What complex numbers have the same absolute value as sqrt(3)+i but subtend a right angle with it at O in the complex plane?

Oct 31, 2016

(2) $\text{ "-1+isqrt(3)" }$ or $\text{ } 1 - i \sqrt{3}$

#### Explanation:

Assuming 'O' is $0$, i.e. the origin, we are basically asking what Complex numbers do you get from $\sqrt{3} + i$ by rotating by a right angle - clockwise or anticlockwise - about $0$.

Rotating anticlockwise about $0$ by a right angle is the same as multiplying by $i = \cos \left(\frac{\pi}{2}\right) + i \sin \left(\frac{\pi}{2}\right)$:

$i \left(\sqrt{3} + i\right) = i \sqrt{3} + {i}^{2} = i \sqrt{3} - 1 = - 1 + i \sqrt{3}$

Rotating clockwise about $0$ by a right angle is the same as multiplying by $- i = \cos \left(- \frac{\pi}{2}\right) + i \sin \left(- \frac{\pi}{2}\right)$:

$- i \left(\sqrt{3} + i\right) = - i \sqrt{3} - {i}^{2} = - i \sqrt{3} + 1 = 1 - i \sqrt{3}$

(2) $\text{ "-1+isqrt(3)" }$ or $\text{ } 1 - i \sqrt{3}$