2.1.29*. Bitta nuqtadan \( \vec{a}(-12 ; 16), \vec{b}(12 ; 5) \) vektorlar o'tkazilgan. \( \vec{a} \) bilan \( \vec{b} \) vektorlar orasidagi burchakni teng ikkiga bo'ladigan va shu nuqtadan chiqqan birlik vektorning koordinatalari topilsin. 2.1.30*. Bitta nuqtadan \( \vec{a}(-3 ; 0 ; 4), \vec{b}(5 ;-2 ;-14) \quad \) vektorlar o'tkazilgan. \( \vec{a}, \vec{b} \) vektorlar orasidagi burchakni teng ikkiga bo'ladigan birlik vektor topilsin.

Keling, berilgan masalalarni bosqichma-bosqich yechamiz. ### Masala 2.1.29 Berilgan vektorlar: - \( \vec{a}(-12, 16) \) - \( \vec{b}(12, 5) \) 1. **Vektorlarning uzunligini hisoblaymiz:** \[ |\vec{a}| = \sqrt{(-12)^2 + 16^2} = \sqrt{144 + 256} = \sqrt{400} = 20 \] \[ |\vec{b}| = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \] 2. **Vektorlarning kosinus burchagini hisoblaymiz:** \[ \cos(\theta) = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| |\vec{b}|} \] \[ \vec{a} \cdot \vec{b} = (-12) \cdot 12 + 16 \cdot 5 = -144 + 80 = -64 \] \[ \cos(\theta) = \frac{-64}{20 \cdot 13} = \frac{-64}{260} = -\frac{32}{130} = -\frac{16}{65} \] 3. **Burchakni topamiz:** \[ \theta = \cos^{-1}\left(-\frac{16}{65}\right) \] 4. **Burchakni teng ikkiga bo'ladigan burchak:** \[ \frac{\theta}{2} \] 5. **Birlik vektorning koordinatalarini topamiz:** Burchakni teng ikkiga bo'lish uchun, \( \vec{a} \) va \( \vec{b} \) vektorlarining birikmasini hisoblaymiz: \[ \vec{u} = \vec{a} + \vec{b} = (-12 + 12, 16 + 5) = (0, 21) \] Birlik vektor: \[ \vec{u}_{\text{birlik}} = \frac{\vec{u}}{|\vec{u}|} \] \[ |\vec{u}| = \sqrt{0^2 + 21^2} = 21 \] \[ \vec{u}_{\text{birlik}} = \left(0, \frac{21}{21}\right) = (0, 1) \] ### Masala 2.1.30 Berilgan vektorlar: - \( \vec{a}(-3, 0, 4) \) - \( \vec{b}(5, -2, -14) \) 1. **Vektorlarning uzunligini hisoblaymiz:** \[ |\vec{a}| = \sqrt{(-3)^2 + 0^2 + 4^2} = \sqrt{9 + 0 + 16} = \sqrt{25} = 5 \] \[ |\vec{b}| = \sqrt{5^2 + (-2)^2 + (-14)^2} = \sqrt{25 + 4 + 196} = \sqrt{225} = 15 \] 2. **Vektorlarning skalar ko'paytmasini hisoblaymiz:** \[ \vec{a} \cdot \vec{b} = (-3) \cdot 5 + 0 \cdot (-2) + 4 \cdot (-14) = -15 + 0 - 56 = -71 \] 3. **Kosinus burchagini hisoblaymiz:** \[ \cos(\theta) = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| |\vec{b}|} = \frac{-71}{5 \cdot 15} = \frac{-71}{75} \] 4. **Burchakni topamiz:** \[ \theta = \cos^{-1}\left(-\frac{71}{75}\right) \] 5. **Burchakni teng ikkiga bo'ladigan burchak:** \[ \frac{\theta}{2} \] 6. **Birlik vektorning koordinatalarini topamiz:** Vektorlar yig'indisi: \[ \vec{u} = \vec{a} + \vec{b} = (-3 + 5, 0 - 2, 4 - 14) = (2, -2, -10) \] Birlik vektor: \[ |\vec{u}| = \sqrt{2^2 + (-2)^2 + (-10)^2} = \sqrt{4 + 4 + 100} = \sqrt{108} = 6\sqrt{3} \] \[ \vec{u}_{\text{birlik}} = \left(\frac{2}{6\sqrt{3}}, \frac{-2}{6\sqrt{3}}, \frac{-10}{6\sqrt{3}}\right) = \left(\frac{1}{3\sqrt{3}}, \frac{-1}{3\sqrt{3}}, \frac{-5}{3\sqrt{3}}\right) \] Natijalar: - 2.1.29 masalasi uchun birlik vektor: \( (0, 1) \) - 2.1.30 masalasi uchun birlik vektor: \( \left(\frac{1}{3\sqrt{3}}, \frac{-1}{3\sqrt{3}}, \frac{-5}{3\sqrt{3}}\right) \)