Question

The area of the triangle formed by the three complex no. $1+i,1+2iand$ $1-i$ is?

• A. $\frac{1}{2}$
• B. $0$
• C. $\sqrt{2}$
• D. $2$

Answer 1

The correct option is B $0$

$1+i,1+2i,1-i$
$=\frac{1}{2}\left|\begin{array}{ccc}1+i& 1-i& 1\\ 1+2i& 1-2i& 1\\ 1-i& 1+i& 1\end{array}\right|$

$=\frac{1}{2}\left\{\right(1+i\left)\right(1-2i-1-i\left)\right\}-(1-i)(1+2i-1+i)+1\left\{\right(1+2i\left)\right(1+i)-(1-i\left)\right(1-2i\left)\right\}$

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

$=\frac{1}{2}\{-6i+6i\}=0$

points are collinear.