1.1 Vektor Algebra — Nazariya

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Kirish

Vektor — bu kattalik (magnitude) va yo'nalishga ega bo'lgan fizik miqdor.

Robotika, dronlar va raketalarda vektorlar hamma joyda:

• 🎯 Kuch vektori — motorning itarish yo'nalishi
• 🎯 Tezlik vektori — dronning harakat yo'nalishi
• 🎯 Joylashuv vektori — robotning 3D fazodagi pozitsiyasi

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1. Vektor Tushunchasi

Skalyar vs Vektor

Skalyar	Vektor
Faqat kattalik	Kattalik + yo'nalish
Massa, temperatura, vaqt	Kuch, tezlik, tezlanish
$m = 5$ kg	$\vec{F} = (3, 4, 0)$ N

Belgilash

Vektorlarni belgilash usullari:

$$\vec{a}, \quad \mathbf{a}, \quad \overrightarrow{AB}$$

Komponentlar orqali (Dekart koordinatalari):

$$\vec{a} = (a_x, a_y, a_z) = a_x\hat{i} + a_y\hat{j} + a_z\hat{k}$$

Bu yerda $\hat{i}, \hat{j}, \hat{k}$ — birlik vektorlar (x, y, z o'qlari bo'ylab).

Vektorning Uzunligi (Magnitudasi)

$$|\vec{a}| = \sqrt{a_x^2 + a_y^2 + a_z^2}$$

Misol

$\vec{a} = (3, 4, 0)$ uchun: $|\vec{a}| = \sqrt{3^2 + 4^2 + 0^2} = \sqrt{25} = 5$

Birlik Vektor

Yo'nalishni saqlab, uzunligi 1 bo'lgan vektor:

$$\hat{a} = \frac{\vec{a}}{|\vec{a}|}$$

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2. Vektor Amallari

Qo'shish va Ayirish

$$\vec{a} + \vec{b} = (a_x + b_x, a_y + b_y, a_z + b_z)$$ $$\vec{a} - \vec{b} = (a_x - b_x, a_y - b_y, a_z - b_z)$$

Geometrik ma'no: Parallelogramm qoidasi

      b
     /|
    / |
   /  |  a + b
  a---+-----→

Skalyarga Ko'paytirish

$$k\vec{a} = (ka_x, ka_y, ka_z)$$

• $k > 0$ — yo'nalish saqlanadi, uzunlik $k$ marta o'zgaradi
• $k < 0$ — yo'nalish teskarisiga aylanadi

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3. Skalyar Ko'paytma (Dot Product)

Ikki vektorning skalyar ko'paytmasi son qaytaradi:

$$\vec{a} \cdot \vec{b} = a_x b_x + a_y b_y + a_z b_z$$

Yoki burchak orqali:

$$\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta$$

Xossalari

1. Kommutativ: $\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}$
2. Distributiv: $\vec{a} \cdot (\vec{b} + \vec{c}) = \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c}$
3. O'zi bilan: $\vec{a} \cdot \vec{a} = |\vec{a}|^2$

Perpendikulyarlik

Agar $\vec{a} \perp \vec{b}$, u holda:

$$\vec{a} \cdot \vec{b} = 0$$

Qo'llanilish

• Ikki vektor orasidagi burchakni topish
• Ish hisoblash: $W = \vec{F} \cdot \vec{d}$
• Proyeksiya hisoblash

Burchakni Topish

$$\cos\theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}$$

Misol

$\vec{a} = (1, 2, 3)$ va $\vec{b} = (4, -5, 6)$ orasidagi burchak:

$\vec{a} \cdot \vec{b} = 1(4) + 2(-5) + 3(6) = 4 - 10 + 18 = 12$ $|\vec{a}| = \sqrt{1 + 4 + 9} = \sqrt{14}$ $|\vec{b}| = \sqrt{16 + 25 + 36} = \sqrt{77}$ $\cos\theta = \frac{12}{\sqrt{14}\sqrt{77}} \approx 0.365$ $\theta \approx 68.6°$

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4. Vektor Ko'paytma (Cross Product)

Ikki vektorning vektor ko'paytmasi yangi vektor qaytaradi:

$$\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_x & a_y & a_z \\ b_x & b_y & b_z \end{vmatrix}$$

Komponentlar:

$$\vec{a} \times \vec{b} = (a_y b_z - a_z b_y, \quad a_z b_x - a_x b_z, \quad a_x b_y - a_y b_x)$$

Xossalari

1. Antikommutativ: $\vec{a} \times \vec{b} = -(\vec{b} \times \vec{a})$
2. Natija perpendikular: $(\vec{a} \times \vec{b}) \perp \vec{a}$ va $(\vec{a} \times \vec{b}) \perp \vec{b}$
3. Uzunligi: $|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin\theta$

O'ng Qo'l Qoidasi

$\vec{a} \times \vec{b}$ yo'nalishini aniqlash: barmoqlarni $\vec{a}$ dan $\vec{b}$ ga burish, bosh barmoq yo'nalishi natija.

Parallellik

Agar $\vec{a} \parallel \vec{b}$, u holda:

$$\vec{a} \times \vec{b} = \vec{0}$$

Qo'llanilish

• Moment hisoblash: $\vec{\tau} = \vec{r} \times \vec{F}$
• Normal vektor topish (tekislikka perpendikular)
• Uchburchak yuzini hisoblash: $S = \frac{1}{2}|\vec{a} \times \vec{b}|$

Misol

$\vec{a} = (2, 3, 4)$ va $\vec{b} = (5, 6, 7)$:

$\vec{a} \times \vec{b} = (3 \cdot 7 - 4 \cdot 6, \quad 4 \cdot 5 - 2 \cdot 7, \quad 2 \cdot 6 - 3 \cdot 5)$ $= (21 - 24, \quad 20 - 14, \quad 12 - 15) = (-3, 6, -3)$

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5. Proyeksiya

$\vec{a}$ ning $\vec{b}$ ustiga proyeksiyasi:

Skalyar proyeksiya

$$\text{proj}_{\vec{b}}\vec{a} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}$$

Vektor proyeksiya

$$\text{proj}_{\vec{b}}\vec{a} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2}\vec{b}$$

Qo'llanilish

Robotikada: kuchning ma'lum yo'nalish bo'yicha komponenti

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6. Koordinata Tizimlari

Dekart (Cartesian)

$$\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$$

Silindr (Cylindrical)

$$\vec{r} = (\rho, \phi, z)$$

O'tish formulalari:

$$x = \rho\cos\phi, \quad y = \rho\sin\phi, \quad z = z$$

Sferik (Spherical)

$$\vec{r} = (r, \theta, \phi)$$

O'tish formulalari:

$$x = r\sin\theta\cos\phi, \quad y = r\sin\theta\sin\phi, \quad z = r\cos\theta$$

Qo'llanilish

• Dekart: Robot qo'llari, CNC
• Silindr: Aylanuvchi tizimlar
• Sferik: Radar, dron kamerasi yo'nalishi

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7. Aralash Ko'paytma (Scalar Triple Product)

Uch vektor uchun:

$$\vec{a} \cdot (\vec{b} \times \vec{c}) = \begin{vmatrix} a_x & a_y & a_z \\ b_x & b_y & b_z \\ c_x & c_y & c_z \end{vmatrix}$$

Geometrik ma'no: Parallelepiped hajmi

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8. Robotikada Qo'llanilish

Misollar

1. Robot qo'lining oxirgi nuqtasi (End Effector):

# Pozitsiya vektori
position = np.array([0.5, 0.3, 0.8])  # metrda

2. Kuch va moment:

$$\vec{\tau} = \vec{r} \times \vec{F}$$

Bu yerda $\vec{r}$ — aylanish markazidan kuchgacha vektor.

3. Dronning tezlik vektori:

$$\vec{v} = v_x\hat{i} + v_y\hat{j} + v_z\hat{k}$$

4. GPS koordinatalari aylanishi: Sferik → Dekart transformatsiya

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Xulosa

Tushuncha	Formula	Natija
Uzunlik	$\|\vec{a}\| = \sqrt{a_x^2 + a_y^2 + a_z^2}$	Skalyar
Dot product	$\vec{a} \cdot \vec{b} = a_xb_x + a_yb_y + a_zb_z$	Skalyar
Cross product	$\vec{a} \times \vec{b}$	Vektor
Burchak	$\cos\theta = \frac{\vec{a} \cdot \vec{b}}{\|\vec{a}\|\|\vec{b}\|}$	Skalyar
Proyeksiya	$\text{proj}_{\vec{b}}\vec{a} = \frac{\vec{a} \cdot \vec{b}}{\|\vec{b}\|^2}\vec{b}$	Vektor

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