## Problem on geometric proofs with vectors

Use vectors and dot products to prove: if the diagonals of a rectangle are perpendicular, then the rectangle is a square.
• ## Solution

We begin by specifying a general rectangle in terms of unknown vectors. We then use those vectors to find expressions for the diagonals. We will then write down what it means for those vectors to be perpendicular.
Because we are given information about the diagonals of the rectangle, we express them as vectors.
Similarly, the vector going from $\mathbf{0}$ to $\mathbf{a} + \mathbf{b}$ is $\mathbf{a} + \mathbf{b}$.
Recall that
Hence, $$(\mathbf{b}-\mathbf{a}) \cdot (\mathbf{a} + \mathbf{b}) = 0$$
Recalling that
we get $$\left | \mathbf{b} \right |^2 = \left | \mathbf{a} \right |^2.$$
Hence $\left | \mathbf{b} \right | = \left | \mathbf{a} \right |$, which proves that the rectangle is a square.