Pythagorean Theorem Vector Proof

Introduction

The Pythagorean Theorem, a cornerstone of Euclidean geometry, establishes a fundamental relationship between the sides of a right-angled triangle: $ a^2 + b^2 = c^2 $, where $ c $ is the length of the hypotenuse. This theorem has profound implications in mathematics, physics, and engineering, underpinning concepts such as vector addition, coordinate geometry, and spatial reasoning. The vector proof of the theorem leverages the properties of vectors and the dot product to demonstrate the equality of squared magnitudes in a geometric context. By representing the sides of a right-angled triangle as vectors in a coordinate system, the proof elegantly avoids reliance on geometric constructions, instead relying on algebraic and vector calculus principles. This approach not only reinforces the theorem’s validity but also illustrates the interplay between vectors and their magnitudes.

Vector Representation and Coordinate System

To facilitate the vector proof, consider a coordinate system where the right-angled triangle is embedded in a two-dimensional plane. Let the triangle have vertices at points $ A $, $ B $, and $ C $, with $ \angle ABC = 90^\circ $. Assign coordinates to these points:

• Let $ A = (0, 0) $, the origin,
• Let $ B = (a, 0) $, placing the leg $ AB $ along the x-axis,
• Let $ C = (0, b) $, placing the leg $ BC $ along the y-axis.

The vectors representing the sides of the triangle are:

• $ \vec{AB} = (a, 0) $,
• $ \vec{BC} = (0, b) $,
• $ \vec{AC} = (a, b) $, the hypotenuse.

The magnitude squared of each vector is:

• $ |\vec{AB}|^2 = a^2 + 0^2 = a^2 $,
• $ |\vec{BC}|^2 = 0^2 + b^2 = b^2 $,
• $ |\vec{AC}|^2 = a^2 + b^2 $.

This setup aligns with the Pythagorean Theorem, as the squared magnitude of the hypotenuse equals the sum of the