Rectangle

Definition

Simple Definition

A quadrilateral where all four angles are equal is called a rectangle.

Linear Algebraic Definition

On a 2D coordinate plane, for two perpendicular vectors $\mathbf{a}$ and $\mathbf{b}$, the set defined as follows is called a rectangle.

$$ R = \left\{ \lambda_{1}\mathbf{a} + \lambda_{2} \mathbf{b} : 0\leq \lambda_{1},\lambda_{2}\leq 1, \quad \mathbf{a} \cdot \mathbf{b} = 0 \right\} $$

Explanation

By definition, if it is a square, then it is a rectangle, and if it is a rectangle, it is a parallelogram and a trapezoid.

Square

$\implies$ Rectangle $\implies$

$\implies$ Rhombus $\implies$

Parallelogram $\implies$ Trapezoid

Adding the condition that all four sides have equal length (rhombus) results in a square.