- Prove that if
and
, then at least one of
is the zero vector. Give a geometric interpretation of this result.
- Prove that if
is a nonzero vector, and if
and
then
.
- Proof. If we have
then
But, since
we know
and
are orthogonal, so
. Therefore, we must have either
or
. Hence, either
or
is the zero vector
- Proof. First,
Then,
But,
and
implies
or
(by part (a)). Since
by hypothesis, we must have
. Hence,