Why is the dot product sometimes called the scalar product?

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

For geometrical vectors in two or three dimensional space, the
**dot product** of two vectors **u** and **v** is:

u · v= |u| |v| cos θ

The dot product is indicated by the dot between the two vectors. Unfortunately, many browsers do not show the dot very clearly. You may see a very tiny dot, or a small black bar. θ is the angle between the two vectors.

Don't write two vectors next to each other like this:
**uv**
when you want the dot product.
Always put a dot between them:
**u · v** .

In 2D the two vectors lie in a plane (of course) and the angle between them is easy to visualize.

In 3D two vectors also lie in a plane embedded within the 3D space, except when the two vectors are co-linear (when they both point in the same direction).

When two vectors are co-linear, the angle between them is zero and so:

u · v= |u||v| cos 0 = |u||v| 1 = |u||v|