Complex numbers z1, z2, z3 are the vertices A, B, C respectively of an isoceles right angled tri

step 1 In this question it is given that if we have an isosceles triangle which is right angled at A an

Complex numbers z1, z2, z3 are the vertices A, B, C...

Complex numbers $\mathrm{z}_{1}, \mathrm{z}_{2}, \mathrm{z}_{3}$ are the vertices $\mathrm{A}, \mathrm{B}, \mathrm{C}$ respectively of an isoceles right angled triangle right angled at $\mathrm{A} .$ Then, $\left(\mathrm{z}_{2}-\mathrm{z}_{3}\right)^{2}$ equals (a) $2\left(z_{1}-z_{2}\right)\left(z_{1}-z_{3}\right)$ (b) $2\left(z_{2}-z_{1}\right)\left(z_{1}-z_{3}\right)$ (c) $3\left(z_{1}+z_{2}+z_{3}\right)$ (d) $\left(z_{2}-z_{1}\right)\left(z_{1}-z_{3}\right)$

Complex numbers $\mathrm{z}_{1}, \mathrm{z}_{2}, \mathrm{z}_{3}$ are the vertices $\mathrm{A}, \mathrm{B}, \mathrm{C}$ respectively of an isoceles right angled triangle right angled at $\mathrm{A} .$ Then, $\left(\mathrm{z}_{2}-\mathrm{z}_{3}\right)^{2}$ equals
(a) $2\left(z_{1}-z_{2}\right)\left(z_{1}-z_{3}\right)$
(b) $2\left(z_{2}-z_{1}\right)\left(z_{1}-z_{3}\right)$
(c) $3\left(z_{1}+z_{2}+z_{3}\right)$
(d) $\left(z_{2}-z_{1}\right)\left(z_{1}-z_{3}\right)$

Key Concepts

Properties of Isosceles Right Triangles

An isosceles right triangle has specific geometric properties, including equal legs and a right angle. Understanding these properties allows one to derive relationships between the sides and angles that can be expressed algebraically using complex numbers, making it possible to connect geometric configurations with algebraic identities.

Rotation Using Complex Multiplication

In the complex plane, multiplication by certain complex numbers, such as the imaginary unit i, corresponds to rotations by specific angles. This property is particularly useful when dealing with rotated figures or right angles, as it provides a convenient algebraic method to represent geometric rotations.

Complex Numbers in Geometry

Complex numbers provide a powerful way to represent points in the plane, where each point is identified with a complex number. This correspondence allows one to apply algebraic techniques to solve geometric problems, bridging algebra and geometry in a seamless manner.

Algebraic Operations with Complex Numbers

The operations of addition, subtraction, and multiplication of complex numbers are essential tools in manipulating expressions that represent geometric entities. These operations enable abstract handling of translations, rotations, and scalings in the plane through straightforward algebraic computations.

Vector Representation in the Complex Plane

Differences between complex numbers can be interpreted as vectors, representing both the magnitude and direction between points. This concept is key in translating geometric problems into algebraic ones, allowing the use of vector algebra in proofs and derivations involving shapes and distances.

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