Scalar (or "dot") Product

The scalar product of two vectors A und B is defined as:

A · B = |A| |B| cos(f_AB)

Here f_AB is the angle between the vectors. This we can also write as the projection of A onto B (which is to say the component of A along B), A||B = |A|·cos(f_AB), multiplied with the lenght of vector B:

A · B = (A||B) |B|

The scalar product is therefore the area that we obtain by multiplying the length of |B| with the component of A parallel to B. This is shown in the above applet. The area is shown in yellow when the scalar product has a positive value, and in pink when it is negative.

Please note:
This also works the other way around, and we can project B on A. We then get:

A · B = (B||A) |A|

All three of the formulas given here for the scalar product are identical.

Hint:
You can use the mouse to drag the vectors by their head and by their tails.