Die nebenstehende Abbildung liefert das 
Resultat für das skalare Produkt $\overrightarrow{v_1}\cdot \overrightarrow{v_2}$
zweier Vektoren $\overrightarrow{v_1}$ und $\overrightarrow{v_2}$ des $\mathbb R^2$
als Ordinatenabschnitt. Ein Beweis der 
dahintersteckenden Konstruktion (von deren
Richtigkeit man sich empirisch anhand beliebig
 vieler Paare $(\overrightarrow{v_1}, \overrightarrow{v_2})$  mehr oder minder über-
zeugt) stellt eine gehörige Herausforderung dar!

Wien, im August 2013.       Dr. Robert Resel, eh.

A = (0, 0)
B = (7, 2)
C = (1, 5)
u = Vector(A, B)
v = Vector(A, C)
a = OrthogonalLine(C, u)
b = Line(A, B)
D = Intersect(a, b)
w = Vector(A, D)
z = Vector(C, D)
α = Angle(C, D, A)
c = OrthogonalLine(B, xAxis)
E = Intersect(c, xAxis)
d = Ray(D, C)
F = Intersect(d, yAxis)
e = Segment(F, C)
G = Point(yAxis)
H = Point(xAxis)
f = Vector(A, H)
g = Vector(A, G)
h = Vector(B, E)
i = Vector(E, B)
k = Circle(A, 1)
I, J = Intersect(k, xAxis)
j = Line(J, i)
l = Segment(A, D)
K = Intersect(l, j)
m = Segment(J, K)
n = Segment(F, B)
p = Vector(F, G)
q = Line(K, n)
r = Segment(A, F)
S_1 = Intersect(r, q)
s = Segment(S_1, K)
t = OrthogonalLine(F, r)
a_1 = OrthogonalLine(S_1, r)
b_1 = OrthogonalLine(A, r)
M = Point(t)
c_1 = Line(M, r)
N = Point(t)
d_1 = Line(N, c_1)
O = Intersect(b_1, c_1)
P = Intersect(a_1, c_1)
Q = Intersect(d_1, b_1)
e_1 = Segment(N, F)
f_1 = Segment(P, S_1)
g_1 = Segment(Q, A)
u_1 = Vector(N, Q)
v_1 = Vector(Q, N)
w_1 = Vector(O, P)
Skalarpodukt = Vector(P, O)
h_1 = Vector(A, G)
i_1 = Segment(A, B)
L = Intersect(i_1, m)
j_1 = Segment(F, L)
k_1 = Line(B, j_1)
R = (1, 0)
T = (0, 1)
l_1 = Line(T, f_1)
m_1 = Segment(A, B)
U = Intersect(m_1, l_1)
n_1 = Line(U, Skalarpodukt)
V = Intersect(n_1, xAxis)
z_1 = Vector(V, U)
Einheitsstrecke = Vector(U, V)
q_1 = Segment(U, F)
r_1 = Line(B, q_1)
S = Intersect(yAxis, r_1)
s_1 = Line(S, e_1)
t_1 = Line(M, yAxis)
a_2 = Line(N, t_1)
Z = Intersect(a_2, s_1)
A_1 = Intersect(t_1, s_1)
b_2 = Line(M, yAxis)
B_1 = Intersect(b_2, g_1)
u_2 = Vector(B_1, M)
v_2 = Vector(M, B_1)
c_2 = Line(Z, yAxis)
C_1 = Intersect(c_2, g_1)
w_2 = Vector(Z, C_1)
Skalarprodukt = Vector(C_1, Z)
d_2 = Segment(Z, S)
e_2 = Segment(M, F)
f_2 = Segment(S, B)
D_1 = Point(yAxis)
E_1 = Point(xAxis)
g_2 = Vector(A, E_1)
h_2 = Vector(A, D_1)