\overrightarrow{u}

\overrightarrow{v}

\overrightarrow{u}\cdot\overrightarrow{v}

=||\overrightarrow{u}|| ||\overrightarrow{v}|| \cos\theta

p_{\overrightarrow{u}}(\overrightarrow{v})

p_{\overrightarrow{u}}(\overrightarrow{v})

=\frac{\overrightarrow{u}\cdot\overrightarrow{v}}{||\overrightarrow{u}||^2}\overrightarrow{u}

p_{\overrightarrow{u}}(\overrightarrow{v})

\alpha \overrightarrow{v}

p_{\overrightarrow{u}}(\alpha\overrightarrow{v})

\alpha=

p_{\overrightarrow{u}}(\alpha\overrightarrow{v})

p_{\overrightarrow{u}}(\alpha \overrightarrow{v}) 

\overrightarrow{u}\cdot\overrightarrow{v}=0

\overrightarrow{u}\cdot\overrightarrow{v}>0

\overrightarrow{u}\cdot\overrightarrow{v}<0

\overrightarrow{v}

\overrightarrow{u}\cdot\overrightarrow{v}=0

\overrightarrow{v}

\overrightarrow{v}

\overrightarrow{u}\cdot\overrightarrow{v}=\alpha

\alpha=

\overrightarrow{u}\cdot\overrightarrow{v}=?

||\overrightarrow{u}||=

p_{\overrightarrow{u}}(\alpha\overrightarrow{v})=\alpha p_{\overrightarrow{u}}(\overrightarrow{v})

p_{\overrightarrow{u}}(\overrightarrow{v}+\overrightarrow{w})

\overrightarrow{w}

p_{\overrightarrow{u}}(\overrightarrow{w})

\overrightarrow{v}+\overrightarrow{w}

p_{\overrightarrow{u}}(\overrightarrow{v}+\overrightarrow{w})=p_{\overrightarrow{u}}(\overrightarrow{v})+p_{\overrightarrow{u}}(
\overrightarrow{w})

p_{\overrightarrow{u}}(\overrightarrow{v}+\overrightarrow{w})

\overrightarrow{u}\cdot(\overrightarrow{v}+\overrightarrow{w})

\overrightarrow{u}\cdot(\alpha\overrightarrow{v})

d_1 = 1 (slider from 0.1 to 5, step 0.1)
A = (-38.26992631323649, 9.47861492737382)
c_1 = Circle(A, d_1)
B = (-36.96565405099798, 9.088163995366116)
W = (-38.73148416536726, 10.088299840765139)
u = Vector(A, B)
p_1 = OrthogonalLine(W, u, xOyPlane)
C_1 = Point(p_1)
C = (-37.812215531761346, 11.788393037106795)
D = (-36.44367102715572, 12.299187816994808)
v = Vector(C, D)
E = Midpoint(A, B)
F = Midpoint(C, D)
c = false
f = Line(A, B)
g = Line(C, D)
G = Intersect(f, g)
δ = Angle(f, g)
text6: "\theta=" + (LaTeX[δ]) + ""
distanceAB = Distance(A, B)
TextAB: Name[A] + (Name[B]) + " = " + distanceAB
distanceCD = Distance(C, D)
TextCD: Name[C] + (Name[D]) + " = " + distanceCD
text7: "||\overrightarrow{u}||=" + (LaTeX[distanceAB]) + ""
text8: "||\overrightarrow{v}||=" + (LaTeX[distanceCD]) + ""
k = 0 (slider from 0 to 1, step 1)
l = 0 (slider from 0 to 1, step 1)
j = 0 (slider from 0 to 2, step 1)
d: (x(u) * x(v)) + (y(u) * y(v))
H = (-35.49918256623072, 9.610286994565454)
H' = Translate(H, u)
h = Line(H, H')
m = OrthogonalLine(C, h)
p = OrthogonalLine(D, h)
I = Intersect(m, h)
J = Intersect(p, h)
w = Vector(I, J)
q = Segment(C, I)
r = Segment(D, J)
α = Angle(C, I, J)
β = Angle(I, J, D)
e = false
n_1 = 0 (slider from 0 to 5, step 1)
s = Line(C, f)
γ = Angle(g, s)
i_1 = 0 (slider from 0 to 3, step 1)
o: d / (x(u)^(2) + y(u)^(2))
j_1 = 0 (slider from 0 to 1, step 1)
M = (-35.880456044737485, 11.31623468114263)
M' = Translate(M, w)
t = Vector(M, M')
N = Midpoint(M, M')
a_1 = 2 (slider from -5 to 5, step 0.1)
O = (-37.936060857465755, 10.356210311734653)
P: O + (a_1 * v)
u_1 = Vector(O, P)
Q = Midpoint(O, P)
f_1 = OrthogonalLine(O, h)
g_1 = Translate(f_1, u_1)
R = Intersect(f_1, h)
S = Intersect(g_1, h)
v_1 = Vector(R, S)
h_1 = Segment(R, O)
k_1 = Segment(S, P)
ε = Angle(O, R, S)
ζ = Angle(R, S, P)
T = Midpoint(R, S)
b_1 = false
l_1 = 1 (slider from 1 to 2, step 1)
U = (-13.37507965460187, 8.301067495613747)
U' = Translate(U, v_1)
w_1 = Vector(U, U')
V = Midpoint(U, U')
m_1 = 0 (slider from 0 to 4, step 1)
text10: "\overrightarrow{u}\cdot\overrightarrow{v}=" + (LaTeX[d]) + ""
K = Midpoint(I, J)
text14: "p_{\overrightarrow{u}}(\overrightarrow{v})=" + (LaTeX[o]) + "\overrightarrow{u}"
a = false
n = -2 (slider from -2 to 3, step 1)
B_1 = Point(p_1)
A_1: C_1
i = Vector(A_1, B_1)
L = Midpoint(A_1, B_1)
Z = Point(p_1)
D_1 = (-35.814059516891405, 12.132057214831136)
b = Vector(C_1, D_1)
E_1 = Midpoint(C_1, D_1)
e_1: (x(b) * x(u)) + (y(b) * y(u))
text26: "\overrightarrow{u}\cdot\overrightarrow{v}=" + (LaTeX[e_1]) + ""
q_1 = Line(A, B)
F_1 = Intersect(p_1, q_1)
r_1 = Circle(F_1, 1, xOyPlane)
G_1, H_1 = Intersect(p_1, r_1)
I_1, J_1 = Intersect(q_1, r_1)
K_1 = Intersect(p_1, q_1)
η = Angle(I_1, F_1, H_1, xOyPlane)
o_1(x, y) = (x(u) * x) + (y(u) * y) > (x(u) * x(W)) + (y(u) * y(W))
s_1(x, y) = (x(u) * x) + (y(u) * y) < (x(u) * x(W)) + (y(u) * y(W))
t_1 = 2.9 (slider from -10 to 10, step 0.1)
L_1: W + (t_1 / (x(u)^(2) + y(u)^(2)) * u)
f_2 = Line(L_1, p_1)
u_2 = Vector(W, L_1)
a_2: t_1 / (x(u)^(2) + y(u)^(2))
text27: "" + (LaTeX[a_2]) + "\overrightarrow{u}"
M_1: W + (0.5 * u_2)
N_1 = Point(f_2)
v_2 = Vector(C_1, N_1)
O_1 = Midpoint(C_1, N_1)
b_2: (x(v_2) * x(u)) + (y(v_2) * y(u))
text29: "\overrightarrow{u}\cdot\overrightarrow{v}=" + (LaTeX[b_2]) + ""
P_1 = Point(p_1)
Q_1 = Translate(P_1, u_2)
R_1 = Midpoint(P_1, Q_1)
w_2 = Vector(P_1, Q_1)
c_2 = false
d_2 = false
n_2 = 0 (slider from 0 to 1, step 1)
S_1 = (-36.7163130612347, 10.677864593630892)
T_1 = (-36.12562899439527, 10.07289358767158)
e_2 = Vector(S_1, T_1)
U_1 = Midpoint(S_1, T_1)
i_2 = 0 (slider from 0 to 4, step 1)
g_2 = OrthogonalLine(S_1, h)
h_2 = OrthogonalLine(T_1, h)
V_1 = Intersect(g_2, h)
W_1 = Intersect(h_2, h)
θ = Angle(W_1, V_1, S_1)
ι = Angle(T_1, W_1, V_1)
j_2 = Segment(S_1, V_1)
k_2 = Segment(T_1, W_1)
l_2 = Vector(V_1, W_1)
Z_1 = Midpoint(V_1, W_1)
D_2 = (-36.08574275675303, 11.80534467582706)
E_2 = Translate(D_2, v)
F_2 = Translate(E_2, e_2)
p_2 = Vector(D_2, F_2)
G_2 = Midpoint(D_2, F_2)
H_2 = (-37.77408469164373, 8.34756852846653)
I_2 = Translate(H_2, w)
J_2 = Translate(I_2, l_2)
q_2 = Vector(H_2, J_2)
K_2 = Midpoint(H_2, J_2)
r_2 = OrthogonalLine(D_2, h)
s_2 = OrthogonalLine(F_2, h)
L_2 = Intersect(r_2, h)
M_2 = Intersect(s_2, h)
N_2 = Midpoint(L_2, M_2)
t_2 = Vector(L_2, M_2)
f_3 = Segment(D_2, L_2)
g_3 = Segment(F_2, M_2)
κ = Angle(D_2, L_2, M_2)
λ = Angle(L_2, M_2, F_2)
A_2: (x(D_1), y(D_1))
m_2 = 0 (slider from 0 to 1, step 0.01)
B_2: ((1 - m_2) * C) + (m_2 * I)
C_2: ((1 - m_2) * D) + (m_2 * J)
u_3 = Vector(B_2, C_2)