Chiziq va tekislik vektorlar bilan

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

To'g'ri chiziqning vektor tenglamasi

$A$ nuqtadan o'tuvchi, $\mathbf{d}$ yo'nalishli chiziq: $\mathbf{r} = \mathbf{a} + t\,\mathbf{d}$

• Ikki chiziq orasidagi burchak — yo'nalish vektorlari orasidagi burchak.
• Kesishish: $\mathbf{a}_1 + s\,\mathbf{d}_1 = \mathbf{a}_2 + t\,\mathbf{d}_2$ dan uchta tenglama tuzib, $s$ va $t$ ni topasiz. Uchinchi tenglama ham bajarilsa, chiziqlar kesishadi; bajarilmasa va parallel ham bo'lmasa — ayqash (skew).

Nuqtadan chiziqqacha perpendikulyar asosi

$P$ nuqta va $\mathbf{r} = \mathbf{a} + t\,\mathbf{d}$ chiziq uchun asos $N$ quyidagi shartdan topiladi: $\overrightarrow{PN}\cdot\mathbf{d} = 0$

🟡 Dekart shakli

$\mathbf{r} = \begin{pmatrix} x_1 \\ y_1 \\ z_1 \end{pmatrix} + t\begin{pmatrix} l \\ m \\ n \end{pmatrix}$ chiziqning Dekart tenglamasi:

$\frac{x - x_1}{l} = \frac{y - y_1}{m} = \frac{z - z_1}{n}$

🟡 Nuqtaning chiziqqa nisbatan aksi

$N$ — $P$ dan chiziqqa tushirilgan perpendikulyar asosi bo'lsa:

$\mathbf{p}' = 2\mathbf{n} - \mathbf{p}$

🔵 Nuqtadan chiziqqacha masofa

$A$ chiziqdagi nuqta, $\mathbf{d}$ yo'nalish vektori bo'lsa:

$d = \frac{|\overrightarrow{AP}\times\mathbf{d}|}{|\mathbf{d}|}$

🔵 Ayqash chiziqlar orasidagi eng qisqa masofa

$d = \frac{|(\mathbf{b} - \mathbf{a})\cdot(\mathbf{d}_1\times\mathbf{d}_2)|}{|\mathbf{d}_1\times\mathbf{d}_2|}$

🔵 Tekislik tenglamasi

$\mathbf{n} = \begin{pmatrix} a \\ b \\ c \end{pmatrix}$ — normal vektor:

$\mathbf{r}\cdot\mathbf{n} = d \qquad ax + by + cz = d$

🔵 Masofa va burchaklar