11.3-bo'lim : Nuqta mahsuloti

1 - 5 muammolar uchun nuqta mahsulotini aniqlang, \(\vec a\centerdot \vec b\).

1. \(\vec a = 9\vec i - 8\vec k\), \(\vec b = \left\langle {3, - 2,1} \right\rangle \)
2. \(\vec a = \left\langle {4, - 1,0,5} \right\rangle \) , \(\vec b = \left\langle {3,0, - 10,6} \right\rangle \)
3. \(\vec a = \vec i - 5\vec j - 2\vec k\) , \(\vec b = - 4\vec i + 2\vec j + 8\vec k\)
4. \(\displaystyle \left\| {\vec a} \right\| = \frac{1}{4}\), \(\displaystyle \left\| {\vec b} \right\| = \frac{9}{4}\) va ikki vektor orasidagi burchak \(\theta = \pi \).
5. \(\displaystyle \left\| {\vec a} \right\| = 24\), \(\left\| {\vec b} \right\| = 9\) va ikki vektor orasidagi burchak \(\displaystyle \theta = \frac{{2\pi }}{7}\).

6 - 8 muammolar uchun ikkita vektor orasidagi burchakni aniqlang.

1. \(\vec p = 9\vec i - \vec j\), \(\vec q = - 3\vec i - 6\vec j\)
2. \(\vec a = \left\langle {4,0, - 3} \right\rangle \), \(\vec b = 2\vec i + 10\vec j - 11\vec k\)
3. \(\vec w = \left\langle {8,3, - 1, - 4} \right\rangle \), \(\vec v = \left\langle { - 1,9,4, - 8} \right\rangle \)

9-12-masalalar uchun ikkita vektor parallel, ortogonal yoki hech biri emasligini aniqlang.

1. \(\vec q = 7\vec i - 14\vec j - 21\vec k\), \(\vec p = \left\langle { - 4,8,12} \right\rangle \)
2. \(\vec u = \left\langle {5,0, - 2} \right\rangle \), \(\vec q = \left\langle {4, - 7,10} \right\rangle \)
3. \(\vec a = 9\vec i - \vec j + 5\vec k\), \(\vec b = - 2\vec i + 7\vec j + \vec k\)
4. \(\vec v = \left\langle { - 1,3,1,5} \right\rangle \), \(\vec w = \left\langle { - 8,3, - 7, - 2} \right\rangle \)
5. \(\vec a\centerdot \vec b = - 6\), \(\left\| {\vec a} \right\| = 4.3\) va \(\vec a\) va \(\vec b\) orasidagi burchak \(\displaystyle \theta = \frac{\pi }{6}\) ekanligini hisobga olsak, \(\vec b\) birlik vektor yoki yo'qligini aniqlaydi.

14 va 15-masalalar uchun ikkita vektor ortogonal bo'ladigan b qiymatini aniqlang.

1. \(\vec u = \left\langle {3, - 1,6} \right\rangle \), \(\vec v = \left\langle {3, - 2b,1} \right\rangle \)
2. \(\vec u = \left\langle {1 - b,4, - 2} \right\rangle \), \(\vec v = \left\langle {b,6,3b} \right\rangle \)
3. berilgan \(\vec a = \vec i + 3\vec j - 2\vec k\) va \(\vec b = - 3\vec i - 4\vec j + 7\vec k\) \({{\mathop{\rm proj}\nolimits} _{\,\vec a}}\,\vec b\) hisoblashadi.
4. berilgan \(\vec a = \vec i + 3\vec j - 2\vec k\) va \(\vec b = - 3\vec i - 4\vec j + 7\vec k\) \({{\mathop{\rm proj}\nolimits} _{\,\vec b}}\,\vec a\) hisoblashadi.
5. berilgan \(\vec p = \left\langle {5, - 2,1} \right\rangle \) va \(\vec q = \left\langle {0,4,8} \right\rangle \) \({{\mathop{\rm proj}\nolimits} _{\,\vec p}}\,\vec q\) hisoblashadi.
6. berilgan \(\vec u = \left\langle {1,3,0, - 2} \right\rangle \) va \(\vec w = \left\langle { - 2,2,4,1} \right\rangle \) \({{\mathop{\rm proj}\nolimits} _{\,\vec w}}\,\vec u\) hisoblashadi.
7. \(\vec r = \left\langle {5,2, - 7} \right\rangle \) uchun yo'nalish kosinuslari va yo'nalish burchaklarini aniqlang.
8. \(\displaystyle \vec r = \left\langle {\frac{1}{2}, - \frac{3}{4},\frac{5}{2}} \right\rangle \) uchun yo'nalish kosinuslari va yo'nalish burchaklarini aniqlang.
9. \(\left( {c\vec v} \right)\centerdot \vec w = \vec v\centerdot \left( {c\vec w} \right)\) xossasini isbotlang.
10. \(\vec v\centerdot \vec w = \vec w\centerdot \vec v\) xossasini isbotlang.
11. \(\vec v\centerdot \vec 0 = 0\) xossasini isbotlang.
12. \(\vec v\centerdot \vec v = {\left\| {\vec v} \right\|^2}\) xossasini isbotlang.