Question

Let \(z_{1}\) and \(z_{2}\) be two distinct complex numbers and let \(z=\left(1-\right.\) t) \(z_{1}+t z_{2}\) for some real number \(\mathrm{t}\) with \(0<\mathrm{t}<1\). If \(\mathrm{Arg}(\mathrm{w})\) denotes the principal argument of a nonzero complex number w, then A) \(\left|z-z_{1}\right|+\left|z-z_{2}\right|=\left|z_{1}-z_{2}\right|\) B) \(\operatorname{Arg}\left(z-z_{1}\right)=\operatorname{Arg}\left(z-z_{2}\right)\) C) \(\left|\begin{array}{cc}z-z_{1} & \bar{z}-\bar{z}_{1} \\ z_{2}-z_{1} & \bar{z}_{2}-\bar{z}_{1}\end{array}\right|=0\) D) \(\operatorname{Arg}\left(z-z_{1}\right)=\operatorname{Arg}\left(z_{2}-z_{1}\right)\)

Short answer

Option D is correct, because the argument of a complex number remains constant along a straight line, and \(z\) lies on the straight line segment between \(z_1\) and \(z_2\).

Step-by-step solution

Assess the Relation Between z and Given Complex Numbers

Analyze the given expression for z, which is a convex combination of two complex numbers: if we assume that \(t\) is a parameter, then \(z\) traces a line segment between \(z_1\) and \(z_2\). The distance from \(z\) to \(z_1\) plus the distance from \(z\) to \(z_2\) should be the total distance between \(z_1\) and \(z_2\).

Validate Option A

To verify option A, calculate \(\left|z-z_1\right|\) and \(\left|z-z_2\right|\) separately and add them. Using the expression for z, we can write \(z - z_1 = -t z_1 + t z_2\) and \(z - z_2 = (1 - t)z_1 - t z_2\). Then we evaluate the sum \(\left|-t z_1 + t z_2\right| + \left|(1 - t)z_1 - t z_2\right|\) and simplify to see if it equals \(\left|z_1 - z_2\right|\).

Verify Option B

For option B, we need to check if the arguments of \(z-z_1\) and \(z-z_2\) are equal. This would imply that the points z, \(z_1\), and \(z_2\) form an isosceles triangle with z at the vertex, which is not possible as z lies on the line segment between \(z_1\) and \(z_2\), and \(z_1\) and \(z_2\) are distinct.

Evaluate Option C

To evaluate option C, calculate the determinant of the matrix and examine if it equals zero. This determinant represents the product of the distances times the sine of the angle between the corresponding vectors; since \(z\) lies on the segment between \(z_1\) and \(z_2\), these vectors are collinear and the determinant should be zero.

Consider Option D

To examine option D, compute the arguments of both \(z - z_1\) and \(z_2 - z_1\). Since \(z\) lies on the line segment connecting \(z_1\) and \(z_2\), the direction of \(z - z_1\) exactly matches the direction of the entire segment \(z_2 - z_1\), indicating the arguments should be the same.

Find the Correct Statement

Going through each option, it is clear that the true statement is option D because the principal argument remains unchanged along a straight line. Option A is incorrect as we would need the path from \(z_1\) to \(z_2\) via z to be a straight line, which it is already. Option B does not hold as explained. Option C is not true since the vectors aren't generally perpendicular (which would make the determinant zero), but they are collinear in this case due to the definition of z.

Key concepts

Principle Argument of Complex Numbers

The principal argument of a complex number represents the angle it makes with the positive real axis on the complex plane. It is denoted by \textbf{Arg}(z) and is usually measured in radians. Unlike the general argument of a complex number, which can have infinitely many values differing by multiples of \textbf{2\(\pi\)}, the principal argument is confined to \textbf{(-\(\pi\), \(\pi\)]}. For a complex number expressed as \(z = x + yi\), where \(x\) and \(y\) are real numbers, the principal argument is the angle \(\theta\) in the polar form \(r(cos\theta + isin\theta)\).

Understanding the principal argument is crucial when working with complex numbers, especially in geometry and trigonometry where rotations and angles come into play. For instance, in the given exercise, the direction of the line segment joining two points on the complex plane is described using the principal argument. The principal argument of the difference of two complex numbers, like \(\text{Arg}(z_2 - z_1)\), gives the angle of the line segment that connects them.

Geometric Interpretation of Complex Numbers

Complex numbers can be visualized on a two-dimensional plane known as the complex plane or Argand plane, where the horizontal axis represents the real part, and the vertical axis the imaginary part of the number. Each complex number corresponds to a unique point on this plane. Geometric interpretations like rotations, scaling, and translations of points can be executed by performing algebraic operations on their respective complex numbers.

For example, adding a fixed complex number to all points (complex numbers) results in a translation, while multiplying by a complex number with absolute value one results in rotation. In the context of the problem provided, \(z\) being a linear combination of \(z_1\) and \(z_2\) implies that it lies on the line segment connecting the points representing \(z_1\) and \(z_2\) on the complex plane, clearly showcasing the relationship between the algebra of complex numbers and their geometric interpretation.

Complex Number Line Segment

A segment in the context of complex numbers refers to the straight path that joins two points on the complex plane. In the exercise, \(z\) is described as a linear combination of \(z_1\) and \(z_2\), which geometrically means it lies somewhere on the line connecting the two points represented by those complex numbers. This relationship is expressed as \(z = (1 - t)z_1 + tz_2\), where \(t\) is a real number between 0 and 1. The value of \(t\) determines the exact location of \(z\) on the line segment; if \(t\) is closer to 0, \(z\) is nearer to \(z_1\), and if \(t\) is closer to 1, it is nearer to \(z_2\).

As shown in the solution, this property is used to understand the distances and arguments between the points, providing insight into the geometric relationships of complex numbers.

Convex Combination of Complex Numbers

A convex combination of two complex numbers is a linear combination where the coefficients sum up to 1 and are non-negative. This concept reflects a weighted average between two points, ensuring that the resulting point, in this case \(z\), is always between \(z_1\) and \(z_2\) on the complex plane. Given by the formula \(z = (1 - t)z_1 + tz_2\) where \(0 \leq t \leq 1\), it implies that no matter the value of \(t\), \(z\) will not exceed the boundaries set by \(z_1\) and \(z_2\).

This is particularly significant in geometry and vector calculus as it allows representations of points that lie within a given range, such as in this exercise where \(z\) is confined to the segment connecting \(z_1\) and \(z_2\). The idea behind the convex combination is highly valued in optimization and mathematical modeling, where it often represents mixtures or combinations of different elements within a specific set of constraints.