## Problem on geometric proofs with vectors

Show that the lines connecting any point on the semicircle of radius 1 to $(1,0)$ and $(-1,0)$ are perpendicular.
• ## Solution

Our strategy will be to express the two line segments as vectors and to show that these vectors are perpendicular.
We let $\bfx$ be any point on the semicircle. The two ends of the semicircle can be expressed in vector form as $\bfi$ and $- \bfi$:
Now we express the vectors from $\pm \bfi$ to $\bfx$. Recall that:
Hence the two line segments are given by the vectors $\mathbf{x} - (-\mathbf{i}) = \mathbf{x} + \mathbf{i}$ and $\mathbf{x} - \mathbf{i}$:
As we would like to show that these vectors are perpendicular, we recall
Recalling that
Hence, we have $$\left( \mathbf{x} - \mathbf{i} \right) \cdot \left( \mathbf{x} + \mathbf{i} \right) = \left | \bfx \right |^2 - 1.$$
Because $\mathbf{x}$ is on the unit semicircle, $\left| \mathbf{x} \right| = 1$, hence $\left( \mathbf{x} - \mathbf{i} \right) \cdot \left( \mathbf{x} + \mathbf{i} \right) = 0$, and the line segments are perpendicular.