This demonstrates how to get the formula for the dot product of two vectors using the geometrical definition and a slew of similar triangles.
Geometrically, if A is a unit vector (length 1) then the dot product of B and A represents the length of the orthogonal projection of B onto the line through A.
Although this can be calculated using lengths and cosines of angles, the formula for the dot product is far, far easier to use. Therefore, getting to that formula as quickly as possible seems a reasonable goal. This demonstration uses similar triangles to show how that formula comes into being.
For details, see the text in the demonstration.