Streamlines, cross product and dot product

Why can't a streamline be defined using the dot product:
http://en.wikipedia.org/wiki/Streamlines,_streaklines,_and_pathlines

Why must they be defined using a cross product? My thinking is that a velocity vector is tangential to a streamline, so why can't we express that using a dot product and setting cosine(theta) = 1?

Are we necessarily dealing with unit vectors? That would give +/-1?

Oops - edited previous post to correct.

I don't know any of this stuff but I just looked at the wikipedia page and basically, if you define it with the cross product, you can write a single equation whereas with the dot product, you would need two i.e. (using wikipedia's notation)

$\frac{d\mathbf{x}_s}{ds}\cdot \mathbf{u}(\mathbf{x}_s) = \mid\frac{d\mathbf{x}_s}{ds}\mid \mid\mathbf{u}(\mathbf{x}_s) \mid$

or

$\frac{d\mathbf{x}_s}{ds}\cdot \mathbf{u}(\mathbf{x}_s) = -\mid\frac{d\mathbf{x}_s}{ds}\mid \mid\mathbf{u}(\mathbf{x}_s) \mid$

which is kinda dopey and impractical.

In general, the cross product is convenient for talking about things being parallel and the dot product is more useful for saying when things are perpendicular.

I don't get why you need two equations?

If $u,v$ are three dimensional vectors then



In other words, the cross product of two vectors is 0 if they are parallel whereas the dot product of two vectors is the product of the lengths of the vectors with a sign corresponding to whether the vectors point in the same or opposite direction.