I read this in a tutorial of a university course :
We note that the vectors V, cV are parallel, and conversely, if two
vectors are parallel (that is, they have the same direction), then one
is a scalar multiple of the other.
Q1. There is an implication in the statement that two vectors are parallel if they are in same direction. Isn't it half right ? I mean in 3D space, two lines could not be in same direction and still be parallel right ?
Q2. If the above statement doesn't hold, then saying
one is a scalar multiple of the other.
is also wrong ?
Best Answer
Parallel vectors on a $K$-Vector space $V$, by definiton, means: $$u \parallel v :\Leftrightarrow \exists \lambda \in K: \lambda \cdot u = v$$ Also, Parallel lines are defined by parallelicity of their respective direction vectors, wich, when fixing $0_V$ as an element of the line, implies equality of the two lines.