Vektorlar: asosiy amallar

⭐ — imtihonda eng ko'p ishlatiladigan formulalar. · 🟢 — masalani tezlashtiradi · 🟡 — ilg'or daraja · 🔵 — chuqur o'rganish (olimpiada darajasi)

Belgilar va shakllar

• Vektor — kattaligi va yo'nalishi bor kattalik; skalyar — faqat kattaligi bor.
• Birlik vektorlar: $\mathbf{i}$, $\mathbf{j}$, $\mathbf{k}$ — mos ravishda $x$, $y$, $z$ yo'nalishlarida.
• Komponentli shakl: $\mathbf{a} = x\mathbf{i} + y\mathbf{j} = \begin{pmatrix} x \\ y \end{pmatrix} \qquad \mathbf{a} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}$
• Kattalik-yo'nalish shakli (2D): $(r, \theta)$. O'tkazish:
  • $(r, \theta) \to$ komponentlar: $x = r\cos\theta$, $\;y = r\sin\theta$
  • komponentlar $\to (r, \theta)$: $r = \sqrt{x^2 + y^2}$, $\;\tan\theta = \dfrac{y}{x}$ (chorakni rasm bilan tekshiring)

Asosiy formulalar

• ⭐ Vektor uzunligi (moduli): $|\mathbf{a}| = \sqrt{x^2 + y^2} \qquad |\mathbf{a}| = \sqrt{x^2 + y^2 + z^2}$
• Teng vektorlar: uzunligi va yo'nalishi bir xil, ya'ni barcha komponentlari teng.
• Nuqtaning radius-vektori (position vector): $\overrightarrow{OP} = \mathbf{p}$
• Amallar komponentlar bo'yicha bajariladi: $k\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} kx \\ ky \end{pmatrix} \qquad \begin{pmatrix} a_1 \\ a_2 \end{pmatrix} \pm \begin{pmatrix} b_1 \\ b_2 \end{pmatrix} = \begin{pmatrix} a_1 \pm b_1 \\ a_2 \pm b_2 \end{pmatrix}$
  • $k\mathbf{a}$ — $\mathbf{a}$ ga parallel, uzunligi $|k|$ marta; $-\mathbf{a}$ — teskari yo'nalishda.
  • Vektorlar yig'indisi — natijaviy vektor (resultant): $\overrightarrow{AB} + \overrightarrow{BC} = \overrightarrow{AC}$.
• ⭐ $\overrightarrow{AB} = \mathbf{b} - \mathbf{a}$
• ⭐ Birlik vektor: $\hat{\mathbf{a}} = \dfrac{\mathbf{a}}{|\mathbf{a}|}$

Misol: $\mathbf{a} = 3\mathbf{i} + 4\mathbf{j}$ bo'lsa, $|\mathbf{a}| = 5$, $\hat{\mathbf{a}} = \frac{3}{5}\mathbf{i} + \frac{4}{5}\mathbf{j}$.

Parallel vektorlar va bir chiziqdagi nuqtalar

• $\mathbf{a}$ va $\mathbf{b}$ parallel ⇔ $\mathbf{a} = k\mathbf{b}$ ($k$ — son)
• $A$, $B$, $C$ bir chiziqda ⇔ $\overrightarrow{AB} = k\,\overrightarrow{BC}$

O'rta nuqta va masofa

• Kesma o'rta nuqtasining radius-vektori: $\mathbf{m} = \dfrac{1}{2}(\mathbf{a} + \mathbf{b})$
• Ikki nuqta orasidagi masofa: $AB = |\mathbf{b} - \mathbf{a}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$

🟢 Kesmani $m : n$ nisbatda bo'luvchi nuqta

$\mathbf{p} = \frac{n\,\mathbf{a} + m\,\mathbf{b}}{m + n}$

🟢 Uchburchakning og'irlik markazi

$\mathbf{g} = \frac{\mathbf{a} + \mathbf{b} + \mathbf{c}}{3}$

🟢 Chiziqli bog'liqmaslik

$\mathbf{a}$ va $\mathbf{b}$ nolga teng emas va parallel emas bo'lsa:

$\lambda\mathbf{a} + \mu\mathbf{b} = \lambda'\mathbf{a} + \mu'\mathbf{b} \;\Rightarrow\; \lambda = \lambda',\; \mu = \mu'$

Geometrik isbotlarda koeffitsientlarni tenglashtirishga imkon beradi.

🟢 Yo'nalish burchagi (2D)

$\mathbf{a} = x\mathbf{i} + y\mathbf{j}$ ning $x$ o'qi bilan burchagi: $\tan\theta = \dfrac{y}{x}$