Multiply two vectors and you can get a plain number out. That number tells you how much of one vector points along the other. Here is the part worth watching: drag the vectors past a right angle and the answer flips from positive to negative, passing through exactly zero on the way.
Drag the tips of A and B, or set them with the sliders — Ax, Ay, Bx, By. Five reveals: Projection of A on B, Projection of B on A, Geometric formula, Algebraic formula, and Sign exploration.
The angle between A and B is always drawn, without you turning anything on. The panel shows |A|, |B|, θ, the dot product itself, and whether it is positive, zero or negative. The projections show how far one vector reaches along the other. Geometric formula writes A · B = |A| |B| cos θ with your numbers in it. Algebraic formula does the same for AₓBₓ + AᵧBᵧ, and shows the components on screen. Sign exploration puts three small pictures side by side: angle under 90°, exactly 90°, and over 90°.
Two dimensions only.
Given two vectors, you should be able to say the sign of their dot product before calculating anything.