u+v

2(u+v)

2u

2v

3(u+v)

3v

3u

1/2u

u/2

v/2

(u+v)/2

-(u+v)

-u

-v

-2u

-2v

-2(u+v)

a \cdot \vec{ v } 

3u/2

3v/2

3(u+v)/2

5u/2

5v/2

5(u+v)/2

-u/2

-v/2

-(u+v)/2

-3u/2

-3v/2

-3(u+v)/2

a\; \cdot

Erstes Distributivgesetz der Vektorrechnung

Verändern Sie die Vektoren     und      durch Ziehen an den Endpunkten der zugehörigen Pfeile und die Zahl  a  mithilfe des Schiebereglers.

a\; \cdot \;( \qquad \qquad  \;\;\; + \;\;\; \;\quad \vec{v} \quad \;\;\;\;)

\vec{u}

a \cdot \vec{u}

+

\vec{u}

\vec{v}

A = Intersect(xAxis, yAxis)
B = (4, 0)
u = Vector(A, B)
C = (1, 3)
v = Vector(A, C)
a = 2 (slider from -2 to 3, step 0.5)
b = Line(B, v)
c = Line(C, u)
D = Intersect(b, c)
w = Vector(A, D)
d = Segment(B, D)
e = Segment(C, D)
f = Line(A, B)
g = Line(A, C)
B'_1 = Translate(B, u)
C'_1 = Translate(C, v)
h = Line(B'_1, g)
i = Line(C'_1, f)
E = Intersect(h, i)
z = Vector(A, E)
j = Segment(C'_1, E)
k = Segment(B'_1, E)
l = Vector(A, B'_1)
m = Vector(A, C'_1)
B'_2 = Mirror(B, B'_1)
C'_2 = Mirror(C, C'_1)
n = Line(B'_2, g)
p = Line(C'_2, f)
H = Intersect(n, p)
q = Vector(A, H)
r = Segment(B'_2, H)
s = Segment(C'_2, H)
t = Vector(A, C'_2)
u_1 = Vector(A, B'_2)
I = Midpoint(A, B)
J = Midpoint(A, C)
v_1 = Vector(A, I)
w_1 = Vector(A, J)
a_1 = Line(I, v)
b_1 = Line(J, u)
F = Intersect(a_1, b_1)
z_1 = Vector(A, F)
c_1 = Line(A, C)
d_1 = Line(B, A)
B' = Mirror(B, A)
C' = Mirror(C, A)
D' = Mirror(D, A)
e_1 = Vector(A, D')
f_1 = Vector(A, B')
g_1 = Vector(A, C')
h_1 = Segment(B', D')
i_1 = Segment(C', D')
A'_1 = Mirror(A, B')
A'_2 = Mirror(A, C')
A' = Mirror(A, D')
j_1 = Vector(A, A'_1)
k_1 = Vector(A, A'_2)
l_1 = Vector(A, A')
m_1 = Segment(A'_1, A')
n_1 = Segment(A'_2, A')
p_1 = Segment(I, F)
q_1 = Segment(J, F)
text1: "\left(\left[\begin{array}{c}" + (x(B)) + " \\ " + (y(B)) + " \\ \end{array} \right]\right."
text3: "=\left[\begin{array}{c}" + (a * x(D)) + " \\ " + (a * y(D)) + " \\ \end{array} \right]"
text9: "=\left[\begin{array}{c}" + (a * x(D)) + " \\ " + (a * y(D)) + " \\ \end{array} \right]"
G = Midpoint(B, B'_1)
K = Midpoint(C, C'_1)
r_1 = Vector(A, G)
s_1 = Vector(A, K)
t_1 = Line(G, s_1)
a_2 = Line(K, r_1)
L = Intersect(t_1, a_2)
u_2 = Vector(A, L)
b_2 = Segment(G, L)
c_2 = Segment(K, L)
K' = Mirror(K, C'_1)
G' = Mirror(G, B'_1)
v_2 = Vector(A, G')
w_2 = Vector(A, K')
d_2 = Line(G', w_2)
e_2 = Line(K', v_2)
M = Intersect(d_2, e_2)
z_2 = Vector(A, M)
f_2 = Segment(G', M)
g_2 = Segment(K', M)
N = Midpoint(A, B')
O = Midpoint(A, C')
P = Midpoint(A, D')
h_2 = Vector(A, N)
i_2 = Vector(A, O)
j_2 = Vector(A, P)
k_2 = Segment(N, P)
l_2 = Segment(O, P)
N' = Mirror(N, B')
O' = Mirror(O, C')
m_2 = Vector(A, N')
n_2 = Vector(A, O')
P' = Mirror(P, D')
p_2 = Vector(A, P')
q_2 = Segment(N', P')
r_2 = Segment(O', P')
text2: "+\!\!\!\!\!\left.\left[\begin{array}{c}" + (x(C)) + " \\ " + (y(C)) + " \\ \end{array} \right]\right)"
text7: "\left[\begin{array}{c}" + (a * x(B)) + " \\ " + (a * y(B)) + " \\ \end{array} \right]"
text8: "+\,\left[\begin{array}{c}" + (a * x(C)) + " \\ " + (a * y(C)) + " \\ \end{array} \right]"