Dario's Blog

Let \(\vec{v}\) be a vector on a plane. Let’s assume that there exist some other vectors \(\vec{i}\) and \(\vec{j}\) on the same plane such that, for some values of \(x, y \in \mathbb{R} \) we can express \(\vec{v}\) as: $$ \vec{v} = x \vec{i} + y \vec{j} $$ In such a scenario, we can call the ordered list \((\vec{i}, \vec{j})\) a coordinate system for the plane and we call the pair \((x, y)\) the coordinates of \(\vec{v}\) in that coordinate system. [Read More]

In this post I will provide an intuitive proof of a very foundational little result in elementary geometry: the Pythagorean theorem. Statement Given a right triangle as the one in Figure 1.i with legs a and b and hypothenuse c, the Pythagorean theorem states that the area of the square whose side is c is the sum of the areas of the two squares having sides a and b. See Figure 1. [Read More]