12.2-bo'lim : Chiziqlar tenglamalari

1 – 4 masalalar uchun chiziq tenglamasini vektor, parametrik va simmetrik shaklda bering.

1. \(\left( {7, - 3,1} \right)\) va \(\left( { - 2,1,4} \right)\) nuqtalari orqali chiziq.
2. \(\left( {1, - 5,0} \right)\) nuqtasi orqali va \(\vec r\left( t \right) = \left\langle {8 - 3t, - 10 + 9t, - 1 - t} \right\rangle \) tomonidan berilgan chiziqqa parallel chiziq.
3. \(\left( {1, - 7,14} \right)\) nuqtasi orqali va \(x = 6t\), \(y = 9\), \(z = 8 - 16t\) tomonidan berilgan chiziqqa parallel chiziq.
4. \(\left( { - 7,2,4} \right)\) nuqtasi orqali chiziq va \(\vec v = \left\langle {0, - 9,1} \right\rangle \) va \(\vec w = 3\vec i + \vec j - 4\vec k\) ga ortogonal.

5-7-masalalar uchun ikkita chiziq parallel, ortogonal yoki hech biri emasligini aniqlang.

1. \(\vec r\left( t \right) = \left\langle {4 - 7t, - 10 + 5t,21 - 4t} \right\rangle \) tomonidan berilgan chiziq va \(\vec r\left( t \right) = \left\langle { - 2 + 3t,7 + 5t,5 + t} \right\rangle \) tomonidan berilgan chiziq.
2. \(\left( {10, - 4,18} \right)\) va \(\left( {5,6, - 7} \right)\) nuqtalari orqali chiziq va \(x = 5 + 3t\), \(y = - 6t\), \(z = 1 + 15t\) tomonidan berilgan chiziq.
3. \(x = 29\), \(y = - 3 - 6t\), \(z = 12 - t\) tomonidan berilgan chiziq va \(\vec r\left( t \right) = \left\langle {12 - 14t,2 + 7t, - 10 + 3t} \right\rangle \) tomonidan berilgan qator.

8 10 masalalar uchun ikkita chiziqning kesishish nuqtasini aniqlang yoki ularning kesishmasligini ko'rsating.

1. \(\left( {0, - 9, - 1} \right)\) va \(\left( {1,6, - 3} \right)\) nuqtalaridan o'tuvchi chiziq va \(\vec r\left( t \right) = \left\langle { - 9 - 4t,10 + 6t,1 - 2t} \right\rangle \) tomonidan berilgan chiziq.
2. \(x = 1 + 6t\), \(y = - 1 - 3t\), \(z = 4 + 12t\) tomonidan berilgan qator va \(x = 4 + t\), \(y = - 10 - 8t\), \(z = 3 - 5t\) tomonidan berilgan qator.
3. \(\vec r\left( t \right) = \left\langle {14 + 5t, - 3t,1 + 7t} \right\rangle \) tomonidan berilgan chiziq va \(\vec r\left( t \right) = \left\langle {3 - 3t,5 + 2t, - 2 + 4t} \right\rangle \) tomonidan berilgan qator.
4. \(\left( { - 5,4, - 1} \right)\) va \(\left( { - 3, - 5,0} \right)\) orqali o'tuvchi chiziq yz-tekislikni kesib o'tadimi? Agar shunday bo'lsa, fikrni bering.
5. \(\vec r\left( t \right) = \left\langle {6 + t, - 8 + 14t,4t} \right\rangle \) tomonidan berilgan chiziq xz tekisligi bilan kesishadimi? Agar shunday bo'lsa, fikrni bering.
6. \(x = 16t\), \(y = - 4 - 9t\), \(z = 34\) tomonidan berilgan chiziq uchta koordinata tekisligidan qaysi biri bilan kesishadi?