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

Then,

which implies has a unique minimum at

*Proof.* We compute the dot product,

Then we have

Therefore,

*Related*