Equivalence of Vector Statements: Proofs and Solutions

[Bipolarity]

Homework Statement

Proof that the following statements are all equivalent. First assume that none of the vectors are zero vectors. Then prove it in the degenerate case, where the vectors are zero vectors.

1) $u = kv$ where k is a scalar.
2) $u \times v = 0$
3) $u \cdot v = ||u|| ||v||$
4) $||u+v|| = ||u|| + ||v||$

Homework Equations

The Attempt at a Solution

In order to prove this, we must show that the truth of each of these statements implies the truth of the other. I was able to show that the truth of the first statement implies the truth of the other three, but have not been able to show the converses. For example, how would I prove that (4) implies (1)? I would need to come up with some scalar k such that u = kv? But how could I generate this scalar?

Any ideas are appreciated.

BiP

 

[LCKurtz]

You don't have to show they all imply 1 directly. What about 4 implies 3 implies 1? Start by squaring 4.