$\vec{a}$

$\vec{b}$

$\vec{b_{\parallel}}$

$\vec{b_{\perp}}$

O: (0 | 0)
K = Point(yAxis)
A = (1, 0)
a = Vector(O, A)
h = Circle(O, K)
B = Point(h)
b = Vector(O, B)
c: (a * b)
d: (abs(a) * abs(b))
Text2: "|\vec{a}| \cdot |\vec{b}| = \sqrt{" + (LaTeX[(x(a))^(2)]) + "+" + (LaTeX[(y(a))^(2)]) + "} \cdot \sqrt{" + (LaTeX[(x(b))^(2)]) + "+" + (LaTeX[(y(b))^(2)]) + "} = " + (LaTeX[d]) + ""
f = Line(O, A)
g = OrthogonalLine(B, f)
C = Intersect(f, g)
u = Vector(O, C)
v = Vector(C, B)
Text1: "\vec{a} \cdot \vec{b} =  \left( \begin{array}{c} " + (LaTeX[x(a)]) + " \\ " + (LaTeX[y(a)]) + " \end{array} \right) \cdot\left( \begin{array}{c} " + (LaTeX[x(b)]) + " \\ " + (LaTeX[y(b)]) + " \end{array} \right) = (" + (LaTeX[x(a)]) + ") \cdot (" + (LaTeX[x(b)]) + ")+ (" + (LaTeX[y(a)]) + ") \cdot (" + (LaTeX[y(b)]) + ") = " + (LaTeX[c]) + ""
e: (a * u)
Text3: "\vec{a} \cdot \vec{b_{\parallel}} =  \left( \begin{array}{c} " + (LaTeX[x(a)]) + " \\ " + (LaTeX[y(a)]) + " \end{array} \right) \cdot\left( \begin{array}{c} " + (LaTeX[x(u)]) + " \\ " + (LaTeX[y(u)]) + " \end{array} \right) = (" + (LaTeX[x(a)]) + ") \cdot (" + (LaTeX[x(u)]) + ")+ (" + (LaTeX[y(a)]) + ") \cdot (" + (LaTeX[y(u)]) + ") = " + (LaTeX[e]) + ""
α = Angle(a, b)
i: cos(α)
j: abs(a)
k: abs(u)
l: (abs(a) * abs(u))
Text4: "cos(\alpha)= " + (LaTeX[i]) + ""
m: abs(b)
Text5: "\underbrace{|\vec{a}|}_{" + (LaTeX[j]) + "} \cdot \underbrace{|\vec{b_{\parallel}}|}_{" + (LaTeX[k]) + "} = " + (LaTeX[l]) + ""
s: ((abs(a) * abs(b)) * i)
w: acosd(i)
Text6: "w(\vec{a},\vec{b}) = acos(\vec{a} \cdot \vec{b}) =" + (LaTeX[w]) + ""