11.3-bo'lim: Nuqta mahsuloti

1-3-masalalar uchun \(\vec a\centerdot \vec b\) nuqta mahsulotini aniqlang.

1. \(\vec a = \left\langle {9,5, - 4,2} \right\rangle \), \(\vec b = \left\langle { - 3, - 2,7, - 1} \right\rangle \) Yechim
2. \(\vec a = \left\langle {0,4, - 2} \right\rangle \) , \(\vec b = 2\vec i - \vec j + 7\vec k\) Yechim
3. \(\left\| {\vec a} \right\| = 5\), \(\displaystyle \left\| {\vec b} \right\| = \frac{3}{7}\) va ikki vektor orasidagi burchak \(\displaystyle \theta = \frac{\pi }{{12}}\). Yechim

4 va 5-masalalar uchun ikkita vektor orasidagi burchakni aniqlang.

1. \(\vec v = \left\langle {1,2,3,4} \right\rangle \), \(\vec w = \left\langle {0, - 1,4, - 2} \right\rangle \) Yechim
2. \(\vec a = \vec i + 3\vec j - 2\vec k\), \(\vec b = \left\langle { - 9,1, - 5} \right\rangle \) Yechim

6-8-masalalar uchun ikkita vektor parallel, ortogonal yoki hech biri emasligini aniqlang.

1. \(\vec q = \left\langle {4, - 2,7} \right\rangle \), \(\vec p = - 3\vec i + \vec j + 2\vec k\) Yechim
2. \(\vec a = \left\langle {3,10} \right\rangle \), \(\vec b = \left\langle {4, - 1} \right\rangle \) Yechim
3. \(\vec w = \vec i + 4\vec j - 2\vec k\), \(\vec v = - 3\vec i - 12\vec j + 6\vec k\) Yechim
4. berilgan \(\vec a = \left\langle { - 8,2} \right\rangle \) va \(\vec b = \left\langle { - 1, - 7} \right\rangle \) \({{\mathop{\rm proj}\nolimits} _{\,\vec a}}\,\vec b\) hisoblashadi. Yechim
5. berilgan \(\vec u = 7\vec i - \vec j + \vec k\) va \(\vec w = - 2\vec i + 5\vec j - 6\vec k\) \({{\mathop{\rm proj}\nolimits} _{\,\vec w}}\,\vec u\) hisoblashadi. Yechim
6. \(\displaystyle \vec r = \left\langle { - 3, - \frac{1}{4},1} \right\rangle \) uchun yo'nalish kosinuslari va yo'nalish burchaklarini aniqlang. Yechim