Question

The complex numbers \(\mathrm{z}_{1}, \mathrm{z}_{2}\) and \(\mathrm{z}_{3}\) satisfying \(\left[\left(z_{1}-z_{3}\right) /\left(z_{2}-z_{3}\right)\right]=[(1-i \sqrt{3}) / 2]\) are the vertices of a triangle which is (a) of area zero (b) right angled isosceles (c) equilateral (d) obtuse-angle isosceles

Short answer

The triangle formed by the complex numbers \(z_1, z_2, z_3\) satisfying the given condition is an equilateral triangle. Hence, the correct option is (c) equilateral.

Step-by-step solution

Simplify the given condition

We are given that: \[\frac{z_1-z_3}{z_2-z_3} = \frac{1-i\sqrt{3}}{2}\] Our first step is to simplify the condition. Let's find the reciprocal of the given complex number. Recall that the reciprocal of a complex number \(a+bi\) is given by: \[\frac{1}{a+bi} = \frac{a-bi}{a^2+b^2}\] Using this, we can find the reciprocal of \(\frac{1-i\sqrt{3}}{2}\): \[\frac{1}{\frac{1-i\sqrt{3}}{2}} = \frac{2}{1-i\sqrt{3}} = \frac{2(1+i\sqrt{3})}{(1-i\sqrt{3})(1+i\sqrt{3})} = 2(1+i\sqrt{3})\] Now, we can rewrite the given condition as: \[(z_1-z_3) = (z_2-z_3) \times 2(1+i\sqrt{3})\]

Interpret geometrically

We can interpret the expression \((z_1-z_3)\) and \((z_2-z_3)\) as the vectors that represent the sides of the triangle. Specifically, they represent the sides going from vertex \(z_3\) to vertices \(z_1\) and \(z_2\), respectively. Now our equation becomes: \[side_{z_3z_1} = side_{z_3z_2} \times 2(1+i\sqrt{3})\] This means that the vector going from vertex \(z_3\) to \(z_1\) is \(2(1+i\sqrt{3})\) times the vector going from vertex \(z_3\) to \(z_2\).

Determine the type of triangle

The relation we obtained in Step 2 suggests that the side from vertex \(z_3\) to \(z_1\) is \(2(1+i\sqrt{3})\) times the side from vertex \(z_3\) to \(z_2\). Notice that \(1+i\sqrt{3}\) represents a \(120^{\circ}\) counterclockwise rotation. So, the side \(z_3z_1\) is \(2\) times in length and is \(120^{\circ}\) rotated compared to the side \(z_3z_2\). Therefore, the triangle is equilateral with sides \(z_3z_1\) and \(z_3z_2\). Since we found the type of triangle, the correct option is (c) equilateral.

Key concepts

Complex Plane Geometry

The complex plane, also known as the Argand plane, is a two-dimensional plane where each point is associated with a complex number. The horizontal axis (real axis) represents the real part of the complex number, while the vertical axis (imaginary axis) represents the imaginary part. This unique configuration allows us to visualize complex numbers and operations on them geometrically.

For example, addition of two complex numbers corresponds to vector addition in the plane. Similarly, multiplication by a complex number of modulus 1 represents a rotation, and if the modulus is different from 1, it includes a scaling factor. These geometric interpretations are particularly valuable in understanding problems involving complex numbers in the form of vectors representing positions or transformations within the plane.

To see these principles in action, consider the exercise where complex numbers are used to define the vertices of a triangle. Utilizing the properties of the complex plane, we can deduce the nature of the triangle, which is a cornerstone to solving such problems.

Vector Representation of Complex Numbers

Complex numbers can be represented as vectors in the complex plane, where the number \( z = a + bi \) corresponds to the vector with the tail at the origin and its head at the point \( (a, b) \) on the plane. This vector has both magnitude, given by the modulus \( |z| = \sqrt{a^2 + b^2} \), and direction, determined by the argument of the complex number — the angle it makes with the positive real axis.

Complex Numbers as Transformation Operators

In the context of triangles and geometry, these vectors can represent sides or transformations. Multiplication by a complex number is a linear transformation that can rotate, stretch, or reflect vectors depending on the complex number used as a multiplier. In our exercise, multiplying by \( 2(1+i\sqrt{3}) \) not only scales the vector but also rotates it by \(120^{\text{\textdegree}}\), which is characteristic of an equilateral triangle's internal angles.

Understanding vector representation of complex numbers and their corresponding transformations is pivotal to visualizing and solving problems such as identifying the nature of triangles in the complex plane.

Properties of Equilateral Triangles

Equilateral triangles are a special class of triangles where all three sides are of equal length and all internal angles are equivalent to \(60^{\text{\textdegree}}\). This symmetry makes them particularly simple to analyze in geometric problems.

An important property of equilateral triangles can be observed in our exercise. When one side of the triangle is scaled and rotated by \(120^{\text{\textdegree}}\) and still produces another side of the triangle, this indicates that the triangle must be equilateral. This property is due to the fact that the internal angles of an equilateral triangle add up to \(360^{\text{\textdegree}}\), with each angle being a third of that total.

Application in Complex Plane

As seen in the exercise, complex numbers can be used to encode not only the positions of the triangle's vertices but also the rotational and scaling relationships between sides. The properties of equilateral triangles then allow for the conclusion that if a side is multiplied by a complex number that represents a \(120^{\text{\textdegree}}\) rotation (and potentially a scaling factor), the resulting geometry corresponds to an equilateral triangle – a beautiful interplay between algebra and geometry.