Prove or disprove:
- If is orthogonal to , then
- If
then is orthogonal to .
- Proof. Let and . Then we have
But, by hypothesis, so . Therefore,
since for all real
- Proof. Suppose otherwise, that . Then, we know that for all we have
for all . So, in particular is must hold for both
Both of these are nonzero since by assumption. But then,
and
But since we cannot have both and . Hence, this is a contradiction, so we must have .
b is false, take for example x=1, and A=kB. The inequality holds without A beeing orthogonal to B (for infinite values of k).
Between the 4th and 5th formula, both side were divided by x. What if x is negative?
In (b)