複數的極式與極座標,直角座標的關係
Author: 藍邦偉老師

$B$

$ 向量的內積 $

$O$

$ \overrightarrow{OA} \cdot \overrightarrow{OB}=|\overrightarrow{OA}|\times |\overrightarrow{OB}|\;cosθ$

$Step=%v$

$ \overrightarrow{OA}\cdot \overrightarrow{OB}=x_1x_2+y_1y_2 $

list1 = Sequence(Circle[(0, 0), r], r, 1, 10)
list4 = Sequence(Sequence[(r * cos(((k - 1) * 30)°), r * sin(((k - 1) * 30)°)), k, 1, 12], r, 1, 16)
B = Point(list4)
A = Point(list4)
list2 = Sequence((cos(((k - 1) * 15)°), sin(((k - 1) * 15)°)), k, 1, 24)
list3 = Sequence(Ray[(0, 0), (cos(((k - 1) * 15)°), sin(((k - 1) * 15)°))], k, 1, 24)
α = 4.1887902047863905 rad (slider from 0 to 6.28318530717959, step 15*°)
β = 2.6179938779914944 rad (slider from 0 to 6.28318530717959, step 15*°)
Q_2: 0.7 * Corner[1] + 0.3 * Corner[2]
Q_3: (x(Q_2), y(Corner[3]))
poly1, a_1, q_2, q_3, b_1 = Polygon(Corner[1], Q_2, Q_3, Corner[4])
θ = Angle(B, (0, 0), A)
r: sqrt(x(A)^(2) + y(A)^(2))
O = (0, 0)
u = Vector(O, A)
v = Vector(O, B)
t_A: "$ A(" + (LaTeX[SurdText[x(A)]]) + " ," + (LaTeX[SurdText[y(A)]]) + ")$"
t_B: "$ B(" + (LaTeX[SurdText[x(B)]]) + "," + (LaTeX[SurdText[y(B)]]) + ") $"
a1: x(A)
a2: y(A)
b1: x(B)
b2: y(B)
s = 0 (slider from 0 to 8, step 1)
a = Segment(O, A)
b = Segment(O, B)
text2: "$ =" + (LaTeX[a]) + " \times" + (LaTeX[b]) + "\times cos\;" + (LaTeX[θ]) + "$"
text3: "$ =" + (LaTeX[a * b]) + " \times" + (LaTeX[SurdText[cos(θ)]]) + "$"
text4: "$ =" + (LaTeX[SurdText[a * b * cos(θ)]]) + " $"
text6: "$ =" + (LaTeX[SurdText[a1]]) + "\cdot" + (LaTeX[If[b1 < 0, "(", ""]]) + "" + (LaTeX[SurdText[b1]]) + "" + (LaTeX[If[b1 < 0, ")", ""]]) + "+" + (LaTeX[If[a2 < 0, "(", ""]]) + " " + (LaTeX[SurdText[a2]]) + "" + (LaTeX[If[a2 < 0, ")", ""]]) + "\cdot" + (LaTeX[If[b2 < 0, "(", ""]]) + "" + (LaTeX[SurdText[b2]]) + "" + (LaTeX[If[b2 < 0, ")", ""]]) + "$"
text7: "$ =" + (LaTeX[SurdText[a1 * b1]]) + "" + (LaTeX[If[a2 * b2 < 0, "-" * SurdText[abs(a2 * b2)], "+" * SurdText[a2 * b2]]]) + "$"
text8: "$ =" + (LaTeX[SurdText[a1 * b1 + a2 * b2]]) + " $"