空間向量的內積與外積
Author: 藍邦偉  老師

x

y

z

單位向量

座標軸

x1

y1

z1

x2

y2

z2

空間向量的內積與外積

$ \overrightarrow{a} =($

,

,

)

$ \overrightarrow{b}=( $

,

,

$ ) $

1

a

b

2

外積：\overrightarrow{a}×\overrightarrow{b}

內積： \overrightarrow{a}\cdot{}\overrightarrow{b}

外積的方向

外積的長度

1

Reset

x

y

z

α = 3.9269908169872414 rad (slider from 0 to 6.283185307179586, step 0.017453292519943295)
β = 1.2217304763960306 rad (slider from 0 to 6.283185307179586, step 0.017453292519943295)
O = (10.62929265874172, 3.7695062636818055)
X_{unit}: O + (cos(α), sin(α) * cos(β))
Y_{unit}: O + (-1 * sin(α), cos(α) * cos(β))
Z_{unit}: O + (0, sin(β))
u = Vector(O, X_{unit})
v = Vector(O, Y_{unit})
w = Vector(O, Z_{unit})
L = 5 (slider from 1 to 10, step 1)
a = Vector(O - L * v, O + L * v)
b = Vector(O - L * w, O + L * w)
z = Vector(O - L * u, O + L * u)
A_1: O + L * u
B_1: O + L * v
Z: O + L * w
c = false
d = true
Q_2: 0.35 * Corner[2] + 0.65 * Corner[1]
Q_3: (x(Q_2), y(Corner[3]))
多邊形1, e, q_2, q_3, f = Polygon(Corner[1], Q_2, Q_3, Corner[4])
g = Segment(Q_2, Q_3)
x1 = 0 (slider from -5 to 5, step 0.1)
y1 = 0 (slider from -5 to 5, step 0.1)
z1 = 0 (slider from -5 to 5, step 0.1)
x2 = 0 (slider from -5 to 5, step 0.1)
y2 = 0 (slider from -5 to 5, step 0.1)
z2 = 0 (slider from -5 to 5, step 0.1)
inputBox1 = Textfield(x1)
inputBox2 = Textfield(y1)
inputBox3 = Textfield(z1)
inputBox4 = Textfield(x2)
inputBox5 = Textfield(y2)
inputBox6 = Textfield(z2)
h = false
P1: O + x1 * u + y1 * v + z1 * w
P2: O + x2 * u + y2 * v + z2 * w
i = Vector(O, P1)
j = Vector(O, P2)
k = false
text12: "=" + (If[x1 * x2 > 0, "", "-"]) + (abs(x1)) + "×" + (abs(x2)) + (If[y1 * y2 < 0, "-", "+"]) + (abs(y1)) + "×" + (abs(y2)) + (If[z1 * z2 < 0, "-", "+"]) + (abs(z1)) + "×" + (abs(z2)) + "=" + (x1 * x2 + y1 * y2 + z1 * z2)
text13: "=\left(\left\vert \begin{array}{rr} " + y1 + "&" + z1 + " \\" + y2 + " &" + z2 + " \\ \end{array}  \right\vert ,\left\vert\begin{array}{rr} " + z1 + " &" + x1 + " \\" + z2 + " &" + x2 + " \\ \end{array}   \right\vert,\left\vert \begin{array}{rr}" + x1 + " &" + y1 + "\\" + x2 + " &" + y2 + " \\ \end{array}  \right\vert     \right)"
text14: "=\left(" + (y1 * z2 - y2 * z1) + "," + (z1 * x2 - z2 * x1) + "," + (x1 * y2 - x2 * y1) + "  \right) "
m = false
poly1, o, p1, n, p2 = Polygon(O, P1, P1 + P2 - O, P2)
x3: y1 * z2 - y2 * z1
y3: z1 * x2 - z2 * x1
z3: x1 * y2 - x2 * y1
P3: O + x3 * u + y3 * v + z3 * w
q = Circle(O, 3)
p = Segment(O, P3)
A = Intersect(q, p)
l = false
p3: sqrt(x3^(2) + y3^(2) + z3^(2))
P4: O + 2 / p3 * P3
r = Vector(O, P4)