z1, z2z3 are 3 complex numbers represented by the points A, B, C in the Argand diagram. I

step 1 In this question it is given that three distinct complex numbers, Z1, Z2 and Z3 are collinear. W

z1, z2z3 are 3 complex numbers represented by the...

$$ \begin{array}{l} \mathrm{z}_{1}, \mathrm{z}_{2} \mathrm{z}_{3} \text { are } 3 \text { complex numbers represented by the points } \mathrm{A}, \mathrm{B}, \mathrm{C} \text { in the Argand diagram. If the points } \mathrm{A}, \mathrm{B}, \mathrm{C} \text { are }\\ \text { collinear, prove that }\left|\begin{array}{lll} \mathrm{z}_{1} & \overline{\mathrm{z}}_{1} & 1 \\ \mathrm{z}_{2} & \overline{\mathrm{z}}_{2} & 1 \\ \mathrm{z}_{3} & \overline{\mathrm{z}}_{3} & 1 \end{array}\right|=0 \end{array} $$

$$
\begin{array}{l}
\mathrm{z}_{1}, \mathrm{z}_{2} \mathrm{z}_{3} \text { are } 3 \text { complex numbers represented by the points } \mathrm{A}, \mathrm{B}, \mathrm{C} \text { in the Argand diagram. If the points } \mathrm{A}, \mathrm{B}, \mathrm{C} \text { are }\\
\text { collinear, prove that }\left|\begin{array}{lll}
\mathrm{z}_{1} & \overline{\mathrm{z}}_{1} & 1 \\
\mathrm{z}_{2} & \overline{\mathrm{z}}_{2} & 1 \\
\mathrm{z}_{3} & \overline{\mathrm{z}}_{3} & 1
\end{array}\right|=0
\end{array}
$$

Key Concepts

Complex Conjugation

Complex conjugation is an operation on complex numbers where the sign of the imaginary part is reversed. This operation preserves the real part while reflecting the imaginary part across the real axis. It is frequently utilized in simplifying expressions and establishing symmetric properties in problems dealing with complex numbers.

Determinants in Geometry

Determinants are algebraic expressions derived from square matrices that can indicate properties such as area, volume, and linear dependence. In geometry, the determinant of a matrix constructed from the coordinates of points can be used to test for collinearity; a zero determinant implies that the points do not span any area, thereby confirming their collinearity.

Complex Numbers

Complex numbers are numbers that have both a real and an imaginary part. They are typically represented in the form a + bi, where 'a' denotes the real component and 'bi' denotes the imaginary component. This representation allows for a wide range of applications in algebra and geometry, linking analytic concepts with geometric interpretations.

Argand Diagram

The Argand diagram is a geometric representation of complex numbers, where each complex number is plotted as a point in the plane with the horizontal axis representing its real part and the vertical axis representing its imaginary part. This visualization facilitates the understanding of complex number operations and geometric relationships such as distance and angle between points.

Collinearity

Collinearity refers to the condition where three or more points lie on a single straight line. In geometric terms, if the points are collinear, the area of the triangle formed by any three of them is zero. This concept is useful in proofs where geometric properties are converted into algebraic conditions.

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