Vektor koʻpaytma

🟢 — masalani tezlashtiradi · 🟡 — ilg'or daraja · 🔵 — chuqur o'rganish (olimpiada darajasi)

🔵 Ta'rif

$\mathbf{a}\times\mathbf{b} = \begin{pmatrix} a_2b_3 - a_3b_2 \\ a_3b_1 - a_1b_3 \\ a_1b_2 - a_2b_1 \end{pmatrix}, \qquad |\mathbf{a}\times\mathbf{b}| = |\mathbf{a}|\,|\mathbf{b}|\sin\theta$

• $\mathbf{a}\times\mathbf{b}$ ikkala vektorga perpendikulyar.
• $\mathbf{a}\times\mathbf{b} = -\,\mathbf{b}\times\mathbf{a}$; $\;\mathbf{a}\times\mathbf{a} = \mathbf{0}$

🔵 Yuzalar

$\text{parallelogramm: } S = |\mathbf{a}\times\mathbf{b}| \qquad \text{uchburchak: } S = \tfrac{1}{2}|\overrightarrow{AB}\times\overrightarrow{AC}|$

🔵 Aralash ko'paytma va hajmlar

$V_{\text{parallelepiped}} = |\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})| \qquad V_{\text{tetraedr}} = \tfrac{1}{6}|\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})|$

• $\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c}) = 0$ ⇔ uchta vektor bir tekislikda yotadi.