Question

The points represented by the complex numbers 1 + i, -2 + 3i, $\frac{5}{3}$ i on the Argand diagram are

• A. vertices of an equilateral triangle
• B. vertices of an isosceles triangle
• C. collinear
• D. none of these

Answer 1

Answer: (C)

collinear

Let $z_1=1+i,z_2=-2+3i$ and $z_3=0+\frac{5}{3}i$
Then $\left|\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|=\left|\begin{array}{ccc}1& 1& 1\\ -2& 3& 1\\ 0& 5/3& 1\end{array}\right|$
$=1(3-\frac{5}{3})+1\left(2\right)+\left(\frac{-10}{3}\right)$
$=\frac{4}{3}+2-\frac{10}{3}=\frac{4+6-10}{3}=0$

Therefore the points are collinear.