The scalar product of two vectors A und B is defined as:
Here fAB 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(fAB), multiplied with the lenght of vector 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.
This also works the other way around, and we can project B on A. We then get:
All three of the formulas given here for the scalar product are identical.
You can use the mouse to drag the vectors by their head and by their tails.