Let and consider the line . Let be an arbitrary point on this line.
- Compute .
- Prove there is exactly one point such that is minimal, and compute the minimum distance.
- Prove that is orthogonal to .
- We compute,
- Proof. From part (a) we know the distance function is . Therefore, the derivative is . This derivative is less than 0 for and greater than 0 for . Therefore, has a minimum at . The distance at this point is
- Proof. First, we have
Hence, and are orthogonal