They are parallel if vector a=k*vector b. From there, you can say their scalar multiples each other
But you cant prove 2 vectors parallel just because their scalar multiples each other
For parallel vectors,
is true. This is what it means for a vector to be a scalar multiple of another.
Your statement is a bit contradictory, you seem to be saying that they are parallel if
, but you can't prove them parallel if they are
That doesn't really make much sense to me.
Note that
does imply that
for when k is greater than 0.
being true doesn't imply that they're parallel vectors. The example you gave in your previous post is an example of this. I think that's what you're getting at, and I think the confusion is whether we're talking about the
vectors a and b, or if we're talking about the
magnitudes of vectors a and b.
For parallel vectors, we are talking about the vectors.