Let be a point not on the line in .
- Consider the function
Prove is a quadratic polynomial in and that there is a unique value of , say , at which this polynomial takes on its minimum.
- Prove that is orthogonal to .
- Proof. First, we compute
But, are all just real scalars (by definition of the dot product); therefore,
for scalars given by
which implies has a unique minimum at
- Proof. We compute the dot product,
Then we have