How can we determine if two vectors are parallel?

Question

What are the minimal number of products like dot cross that can give us information if two vectors are parallel ?

What can we say if V*W = 1 assuming V and W are not unit vectors.

Answer 1

So the cross product of two vectors is actually enough. $|V \times W| = 0$ is the condition you want, for the equality $|V \times W| = |V|\cdot |W|\cdot Sin(\theta)$ where $\theta$ is the angle between V and W tells you that they are parallel.

Answer 2

Using Cauchy-Schwarz (assuming we are talking about a Hilbert space, etc...) , $(V\cdot W)^2=V^2W^2$ iff $V$ and $W$ are parallel. I count 3 dot products, so the solution involving 1 cross product is more efficient in this sense, but the cross product is a bit more involved.

If $(V\cdot W)=1$ (my interpretation of your question) and $V^2,W^2\ne1$, then at least one of them has to have norm greater than 1. They could be non parallel or parallel though. But if you require that $V^2,W^2>1$, then they are definitely non-parallel.