Problem on geometric proofs with vectors - Leading Lesson

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.
Begin a geometric proof by labeling important points

First we specify the rectangle with vectors. We place the lower left corner at the origin. Let the top left and bottom right corners be given by the vectors $\mathbf{a}$ and $\mathbf{b}$.
Geometric view of vector addition

By vector addition, the top right corner is given by $\mathbf{a} + \mathbf{b}$.
Because we are given information about the diagonals of the rectangle, we express them as vectors.
Subtraction gives the vector between two points

The vector going from $\mathbf{a}$ to $\mathbf{b}$ is $\mathbf{b} - \mathbf{a}$.
Similarly, the vector going from $\mathbf{0}$ to $\mathbf{a} + \mathbf{b}$ is $\mathbf{a} + \mathbf{b}$.
Recall that
Dot product of perpendicular vectors is zero

If two vectors are perpendicular, their dot product is zero.
Hence, $$(\mathbf{b}-\mathbf{a}) \cdot (\mathbf{a} + \mathbf{b}) = 0$$
Algebra of dot products

Because the dot product is distributive, we expand this equation to \begin{align}
\mathbf{b} \cdot \mathbf{a} + \mathbf{b} \cdot \mathbf{b} - \mathbf{a} \cdot \mathbf{a} - \mathbf{a} \cdot \mathbf{b} &= 0.\\
\end{align}
Algebra of dot products

Because $\mathbf{a}\cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}$, we can simplify this equation to $$\mathbf{b} \cdot \mathbf{b} = \mathbf{a} \cdot \mathbf{a}.$$
Recalling that
Dot product and vector length

The dot product of a vector with itself gives the squared length of that vector.
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.

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