1. There is no one multiplication for vectors. You can define multiplication-like operations; some give scalars (dot product and other inner products) and some give vectors (cross product). Nicely, these behave like regular products when it comes to vector-valued functions: the product rule applies when you differentiate (though you must maintain ordering with cross product!).

2. Whenever we want to isolate the direction of something, we use a unit vector.

3. While it is useful to think of linearly independent sets as “small” and spanning sets as “large”, be careful not to try to apply the converse. {0} is very small, but linearly dependent. {v, 2v, 3v, 4v, 5v, …} is very large, but spans only a dimension 1 space.