Complex Numbers
Geometry of Complex Numbers
Premium Question
Grade 11
Question:
Three points $z_1, z_2, z_3$ are collinear if and only if:
(1) $\frac{z_3 - z_1}{z_2 - z_1}$ is real
(2) $\frac{z_3 - z_1}{z_2 - z_1}$ is purely imaginary
(3) $|z_3 - z_1| = |z_3 - z_2| + |z_2 - z_1|$
(4) Both (1) and (3)
Step-by-Step Solution
Key Concept: Two vectors are parallel iff their ratio is real. $z_3 - z_1$ and $z_2 - z_1$ point in the same direction iff $(z_3 - z_1)/(z_2 - z_1) \in \mathbb{R}$. Also check when option (3) holds geometrically.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (4)