Skalyar koʻpaytma

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

Skalyar ko'paytma va burchak

• ⭐ Skalyar ko'paytma: $\mathbf{a}\cdot\mathbf{b} = a_1b_1 + a_2b_2 \;(\text{2D}) \qquad \mathbf{a}\cdot\mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \;(\text{3D})$
• ⭐ Ikki vektor orasidagi burchak: $\cos\theta = \frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}|\,|\mathbf{b}|}$
• ⭐ Perpendikulyar vektorlar: $\mathbf{a}\cdot\mathbf{b} = 0$

Skalyar ko'paytma xossalari

$\mathbf{a}\cdot\mathbf{b} = \mathbf{b}\cdot\mathbf{a} \qquad \mathbf{a}\cdot(\mathbf{b} + \mathbf{c}) = \mathbf{a}\cdot\mathbf{b} + \mathbf{a}\cdot\mathbf{c} \qquad \mathbf{a}\cdot\mathbf{a} = |\mathbf{a}|^2$

🟢 Uzunlik va ko'paytma

$|\mathbf{a} + \mathbf{b}|^2 = |\mathbf{a}|^2 + 2\,\mathbf{a}\cdot\mathbf{b} + |\mathbf{b}|^2$

Bu vektor shaklidagi kosinuslar teoremasi.

🔵 Tengsizliklar

$|\mathbf{a}\cdot\mathbf{b}| \le |\mathbf{a}|\,|\mathbf{b}| \;\;\text{(Koshi–Bunyakovskiy)} \qquad |\mathbf{a} + \mathbf{b}| \le |\mathbf{a}| + |\mathbf{b}| \;\;\text{(uchburchak)}$

🟡 Proyeksiyalar

$\text{skalyar proyeksiya: } \frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{b}|} \qquad \text{vektor proyeksiya: } \frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{b}|^2}\,\mathbf{b}$

🔵 Yo'nalish kosinuslari

$\mathbf{a} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k}$ uchun:

$\cos\alpha = \frac{x}{|\mathbf{a}|} \quad \cos\beta = \frac{y}{|\mathbf{a}|} \quad \cos\gamma = \frac{z}{|\mathbf{a}|} \qquad \cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1$

💡 AI bilan bog'liqlik

Burchak formulasi $\cos\theta = \dfrac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}|\,|\mathbf{b}|}$ — bu aynan cosine similarity. Matn embeddinglari o'rtasidagi o'xshashlik shu formula bilan o'lchanadi.