2D 座標架
Author: 藍邦偉  老師

$ O $

$ \vec{a} =(x_1,y_1),\vec{b}=(x_2,y_2)$
$⇒\vec{a}\cdot \vec{b}=x_1x_2+y_1y_2$

$ \vec{a}× \vec{c}= \left\vert \begin{array}{cc} x_1 & y_1 \- y_2 & x_2 \ \end{array}  \right\vert $
$\;\;\;\;\;\;\;\;\;=x_1x_2+y_1y_2=\vec{a}\cdot \vec{b} $

$ 將\;\vec{b}\;旋轉90°，\;得： $
$\vec{c}=\left [\begin{array}{cc} cos90° & -sin90⁰ \ sin90° & cos90° \ \end{array}   \right ] \left [  \begin{array}{cc} x_2  \ y_2 \ \end{array} \right ]$
$\;\;\;=\left [\begin{array}{cc} 0 & -1 \ 1 & 0 \ \end{array}   \right ] \left [  \begin{array}{cc} x_2  \ y_2 \ \end{array} \right ] $
$\;\;\;=(-y_2,\;x_2)$

$ 向量內積的幾何意義 $

$Step=%v$

$ A $

$ B $

$ C $

C'

$ \vec{a} $

$ \vec{b} $

$ \vec{c} $

Reset

u = Vector((x(Corner[1]), 0), (x(Corner[3]), 0))
v = Vector((0, y(Corner[1])), (0, y(Corner[3])))
O = (0, 0)
Q_2: 0.65 * Corner[1] + 0.35 * Corner[2]
Q_3: (x(Q_2), y(Corner[4]))
poly1, q_1, q_2, q_4, q_3 = Polygon(Corner[1], Q_2, Q_3, Corner[4])
A = (0.8726949736241308, 1.64599729066815)
B = (4.921490385666649, 1.8178801147642945)
a = Vector(A, B)
C = (2.725209855549245, 4.548907208736369)
b = Vector(A, C)
α = 0 rad (slider from 0*° to 90*°, step 0.1*°)
C' = Rotate(C, α, A)
w = Vector(A, C')
D: B + C' - A
poly2, a_1, b_1, d, c' = Polygon(A, B, D, C')
s = 0 (slider from 0 to 4, step 1)
c = If(s ≟ 0, α = 0°)
β = Angle(C, A, C')
t1 = 0 (slider from 0 to 1, step 0.01)
B_1: A + t1 * Vector[A, B]
C_1: A + t1 * Vector[A, C]
t3 = 0 (slider from 0 to 1, step 0.01)
z_1 = Vector(A, B_1)
z_2 = Vector(A, C_1)
D_1: B + t3 * Vector[B, D]
D_2: C' + t3 * Vector[C', D]
e = Vector(C', D_2)
f = Vector(B, D_1)
g: s ≟ 0