Question

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?

Answer 1

(2) `" "-1+isqrt(3)" "` or `" "1-isqrt(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(pi/2)+i sin(pi/2)`:

`i(sqrt(3)+i) = isqrt(3)+i^2 = isqrt(3)-1 = -1+isqrt(3)`

Rotating clockwise about `0` by a right angle is the same as multiplying by `-i = cos(-pi/2)+i sin(-pi/2)`:

`-i(sqrt(3)+i) = -isqrt(3)-i^2 = -isqrt(3)+1 = 1-isqrt(3)`

So the answer is:

(2) `" "-1+isqrt(3)" "` or `" "1-isqrt(3)`