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,