Why is the dot product sometimes called the scalar product?

Because it takes two vectors and produces a *scalar* (a real number.)

The angle between two vectors **u** and **v** is
the angle θ that satisfies:

0 <= θ <= 180°

This definition works for both 2D space and 3D space.

The angle is the smallest angle that one vector can be rotated until it aligns with the other.

If two vectors point in opposite directions, what is the angle between them?