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Chapter 7: Problem 51

Equilateral triangles in the complex plane: \(u^{2}+v^{2}+w^{2}=u v+u w+v w\) If the line segments connecting the complex numbers \(u, v,\) and \(w\) form the vertices of an equilateral triangle, the formula shown above holds true. Verify that \(u=2+\sqrt{3} i, v=10+\sqrt{3} i,\) and \(w=6+5 \sqrt{3} i\) form the vertices of an equilateral triangle using the distance formula, then verify the formula given.

Short Answer

Expert verified
The points form an equilateral triangle; the given formula holds true.

Step by step solution

01

Verify Side Lengths

To check if the triangle is equilateral, we need to calculate the distances between each pair of points: \ 1. The distance between \(u\) and \(v\) is: \[ |v - u| = |(10 + \sqrt{3}i) - (2 + \sqrt{3}i)| = |8| = 8 \]2. The distance between \(v\) and \(w\) is: \[ |w - v| = |(6 + 5\sqrt{3}i) - (10 + \sqrt{3}i)| = |(-4 + 4\sqrt{3}i)| \] Calculate \(|-4 + 4\sqrt{3}i|\): \[ \sqrt{(-4)^2 + (4\sqrt{3})^2} = \sqrt{16 + 48} = \sqrt{64} = 8 \]3. The distance between \(u\) and \(w\) is: \[ |w - u| = |(6 + 5\sqrt{3}i) - (2 + \sqrt{3}i)| = |4 + 4\sqrt{3}i| \] Calculate \(|4 + 4\sqrt{3}i|\): \[ \sqrt{4^2 + (4\sqrt{3})^2} = \sqrt{16 + 48} = \sqrt{64} = 8 \]Since all sides have equal length (8), it forms an equilateral triangle.
02

Verify Given Formula

Now, verify the formula \(u^2 + v^2 + w^2 = uv + uw + vw\) using the given vertices \(u, v, w\). Compute each part:1. Calculate \(u^2, v^2, w^2\): - \(u^2 = (2+\sqrt{3}i)^2 = 4 + 4\sqrt{3}i - 3 = 1 + 4\sqrt{3}i\) - \(v^2 = (10+\sqrt{3}i)^2 = 100 + 20\sqrt{3}i - 3 = 97 + 20\sqrt{3}i\) - \(w^2 = (6+5\sqrt{3}i)^2 = 36 + 60\sqrt{3}i - 75 = -39 + 60\sqrt{3}i\)2. Calculate \(uv, uw, vw\): - \(uv = (2+\sqrt{3}i)(10+\sqrt{3}i) = 20 + 2\sqrt{3}i + 10\sqrt{3}i - 3 = 17 + 12\sqrt{3}i\) - \(uw = (2+\sqrt{3}i)(6+5\sqrt{3}i) = 12 + 10\sqrt{3}i + 6\sqrt{3}i - 15 = -3 + 16\sqrt{3}i\) - \(vw = (10+\sqrt{3}i)(6+5\sqrt{3}i) = 60 + 50\sqrt{3}i + 6\sqrt{3}i - 15 = 45 + 56\sqrt{3}i\)3. Check if \(u^2 + v^2 + w^2 = uv + uw + vw\): - \(u^2 + v^2 + w^2 = (1 + 4\sqrt{3}i) + (97 + 20\sqrt{3}i) + (-39 + 60\sqrt{3}i) = 59 + 84\sqrt{3}i\) - \(uv + uw + vw = (17 + 12\sqrt{3}i) + (-3 + 16\sqrt{3}i) + (45 + 56\sqrt{3}i) = 59 + 84\sqrt{3}i\)The equality holds, confirming the points form the vertices of an equilateral triangle.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Complex Numbers
Complex numbers are a fundamental concept in mathematics, particularly in the field of complex plane geometry. They are composed of two parts: a real part and an imaginary part. For example, numbers like \(u = 2 + \sqrt{3}i\) have a real part of 2 and an imaginary part of \(\sqrt{3}\). These numbers are incredibly useful in various mathematical problems because they allow for the representation of two-dimensional points.
  • They have the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit satisfying \(i^2 = -1\).
  • The real part is denoted as \(\text{Re}(z)\), and the imaginary part as \(\text{Im}(z)\).
Understanding how to handle complex numbers is crucial for dealing with geometry problems, especially when these numbers are employed to verify conditions like the vertices of an equilateral triangle.
Distance Formula
The distance formula in the complex plane is an adaptation of the Euclidean distance formula. It helps in calculating the distance between two points represented by complex numbers. In the given problem, we use this formula to confirm that the triangle formed by the complex numbers \(u\), \(v\), and \(w\) is equilateral.The formula is given by:\[|z_2 - z_1| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]where \(z_1 = x_1 + y_1i\) and \(z_2 = x_2 + y_2i\). This effectively means calculating the magnitude of the difference between the two complex numbers.By computing the distances between \(u\), \(v\), and \(w\), and verifying their equality (all distances are 8), we can assert that these points are the vertices of an equilateral triangle.
Geometric Proof
A geometric proof involves demonstrating certain properties or relationships using a step-by-step logical argument. In the context of this exercise, a geometric proof shows that the points \(u\), \(v\), and \(w\) form an equilateral triangle and satisfy the condition \(u^2 + v^2 + w^2 = uv + uw + vw\).Such proofs often require:
  • Verification of fundamental properties, like equal side lengths, using precise calculations.
  • Logical deduction based on known geometric principles presented in a clear, sequential manner.
The provided solution completes this proof by calculating the distances to confirm the triangle's equilateral nature and then validating the given algebraic condition. This method confirms that geometric principles align with algebraic relationships in complex numbers.
Complex Plane Geometry
Complex plane geometry is a fascinating branch of mathematics combining complex numbers and geometric concepts. The complex plane allows us to visualize and solve geometric problems in a two-dimensional space using algebraic methods. In this context, complex numbers "live" on a plane where the real part of the number is the x-coordinate and the imaginary part is the y-coordinate.
  • It enables the translation of geometric problems into algebraic problems, which are often easier to solve.
  • Understanding this geometry involves grasping both the arithmetic of complex numbers and basic geometric principles.
This exercise perfectly illustrates complex plane geometry by equating algebraic and geometric properties for verifying the given shape is an equilateral triangle. It emphasizes the seamless interaction between algebra and geometry facilitated by the complex plane.

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