выпишем координаты данных векторов:
![\vec{a}=(-1,0,5);\ \vec{b}=(-3,2,2);\ \vec{c}=(-2,-4,1)](https://tex.z-dn.net/?f=%5Cvec%7Ba%7D%3D%28-1%2C0%2C5%29%3B%5C%20%5Cvec%7Bb%7D%3D%28-3%2C2%2C2%29%3B%5C%20%5Cvec%7Bc%7D%3D%28-2%2C-4%2C1%29)
a)
координаты:
![3*\vec{a}=(3*(-1),3*0,3*5)=(-3,0,15)\\</p><p>2*\vec{b}=(-6,4,4)](https://tex.z-dn.net/?f=3%2A%5Cvec%7Ba%7D%3D%283%2A%28-1%29%2C3%2A0%2C3%2A5%29%3D%28-3%2C0%2C15%29%5C%5C%3C%2Fp%3E%3Cp%3E2%2A%5Cvec%7Bb%7D%3D%28-6%2C4%2C4%29)
скалярное произведение векторов - число:
![3\vec{a}*2\vec{b}=(-3)*(-6)+0*4+15*4=18+60=78](https://tex.z-dn.net/?f=3%5Cvec%7Ba%7D%2A2%5Cvec%7Bb%7D%3D%28-3%29%2A%28-6%29%2B0%2A4%2B15%2A4%3D18%2B60%3D78)
б)
координаты:
![7*\vec{a}=(-7,0,35)\\</p><p>(-3)*\vec{c}=(6,12,-3)](https://tex.z-dn.net/?f=7%2A%5Cvec%7Ba%7D%3D%28-7%2C0%2C35%29%5C%5C%3C%2Fp%3E%3Cp%3E%28-3%29%2A%5Cvec%7Bc%7D%3D%286%2C12%2C-3%29)
векторное произведение векторов - вектор, находим его координаты:
![7\vec{a}\times (-3\vec{b})=\left|</p><p>\begin{array}{ccc}</p><p>\vec{i} & \vec{j} & \vec{k} \\</p><p>-7 & 0 & 35 \\</p><p>6 & 12 & -3</p><p>\end{array}</p><p>\right|=\vec{i}*\left|</p><p>\begin{array}{cc}</p><p>0 & 35 \\</p><p>12 & -3</p><p>\end{array}</p><p>\right|-\vec{j}*\left|</p><p>\begin{array}{cc}</p><p>-7 & 35 \\</p><p>6 & -3</p><p>\end{array}</p><p>\right|+\vec{k}*\left|</p><p>\begin{array}{cc}</p><p>-7 & 0 \\</p><p>6 & 12</p><p>\end{array}</p><p>\right|=\vec{i}*(-12*35)-\vec{j}*(21-6*35)+\vec{k}*(12*(-7))=\\=-420\vec{i}+189\vec{j}-84*\vec{k}=(-420,189,-84)](https://tex.z-dn.net/?f=7%5Cvec%7Ba%7D%5Ctimes%20%28-3%5Cvec%7Bb%7D%29%3D%5Cleft%7C%3C%2Fp%3E%3Cp%3E%5Cbegin%7Barray%7D%7Bccc%7D%3C%2Fp%3E%3Cp%3E%5Cvec%7Bi%7D%20%26%20%5Cvec%7Bj%7D%20%26%20%5Cvec%7Bk%7D%20%5C%5C%3C%2Fp%3E%3Cp%3E-7%20%26%200%20%26%2035%20%5C%5C%3C%2Fp%3E%3Cp%3E6%20%26%2012%20%26%20-3%3C%2Fp%3E%3Cp%3E%5Cend%7Barray%7D%3C%2Fp%3E%3Cp%3E%5Cright%7C%3D%5Cvec%7Bi%7D%2A%5Cleft%7C%3C%2Fp%3E%3Cp%3E%5Cbegin%7Barray%7D%7Bcc%7D%3C%2Fp%3E%3Cp%3E0%20%26%2035%20%5C%5C%3C%2Fp%3E%3Cp%3E12%20%26%20-3%3C%2Fp%3E%3Cp%3E%5Cend%7Barray%7D%3C%2Fp%3E%3Cp%3E%5Cright%7C-%5Cvec%7Bj%7D%2A%5Cleft%7C%3C%2Fp%3E%3Cp%3E%5Cbegin%7Barray%7D%7Bcc%7D%3C%2Fp%3E%3Cp%3E-7%20%26%2035%20%5C%5C%3C%2Fp%3E%3Cp%3E6%20%26%20-3%3C%2Fp%3E%3Cp%3E%5Cend%7Barray%7D%3C%2Fp%3E%3Cp%3E%5Cright%7C%2B%5Cvec%7Bk%7D%2A%5Cleft%7C%3C%2Fp%3E%3Cp%3E%5Cbegin%7Barray%7D%7Bcc%7D%3C%2Fp%3E%3Cp%3E-7%20%26%200%20%5C%5C%3C%2Fp%3E%3Cp%3E6%20%26%2012%3C%2Fp%3E%3Cp%3E%5Cend%7Barray%7D%3C%2Fp%3E%3Cp%3E%5Cright%7C%3D%5Cvec%7Bi%7D%2A%28-12%2A35%29-%5Cvec%7Bj%7D%2A%2821-6%2A35%29%2B%5Cvec%7Bk%7D%2A%2812%2A%28-7%29%29%3D%5C%5C%3D-420%5Cvec%7Bi%7D%2B189%5Cvec%7Bj%7D-84%2A%5Cvec%7Bk%7D%3D%28-420%2C189%2C-84%29)
находим модуль(длину) полученного вектора:
![|7\vec{a}\times (-3\vec{b})|=\sqrt{420^2+189^2+84^2}=\sqrt{21^2(20^2+9^2+4^2)}=21\sqrt{497}](https://tex.z-dn.net/?f=%7C7%5Cvec%7Ba%7D%5Ctimes%20%28-3%5Cvec%7Bb%7D%29%7C%3D%5Csqrt%7B420%5E2%2B189%5E2%2B84%5E2%7D%3D%5Csqrt%7B21%5E2%2820%5E2%2B9%5E2%2B4%5E2%29%7D%3D21%5Csqrt%7B497%7D)
в)
координаты:
![3\vec{a}=(-3,0,15)\\</p><p>-4\vec{b}=(12,-8,-8)\\</p><p>2\vec{c}=(-4,-8,2)](https://tex.z-dn.net/?f=3%5Cvec%7Ba%7D%3D%28-3%2C0%2C15%29%5C%5C%3C%2Fp%3E%3Cp%3E-4%5Cvec%7Bb%7D%3D%2812%2C-8%2C-8%29%5C%5C%3C%2Fp%3E%3Cp%3E2%5Cvec%7Bc%7D%3D%28-4%2C-8%2C2%29)
смешанное произведение векторов - число, находим его:
![(3\vec{a},(-4\vec{b}),2\vec{c})=\left|</p><p>\begin{array}{ccc}</p><p>-3 & 0 & 15 \\</p><p>12 & -8 & -8 \\</p><p>-4 & -8 & 2</p><p>\end{array}</p><p>\right|=\\=-3*\left|</p><p>\begin{array}{cc}</p><p>-8 & -8 \\</p><p>-8 & 2</p><p>\end{array}</p><p>\right|+15*\left|</p><p>\begin{array}{cc}</p><p>12 & -8 \\</p><p>-4 & -8</p><p>\end{array}</p><p>\right|=-3(-16-64)+15(-96-32)=240-1920=-1680](https://tex.z-dn.net/?f=%283%5Cvec%7Ba%7D%2C%28-4%5Cvec%7Bb%7D%29%2C2%5Cvec%7Bc%7D%29%3D%5Cleft%7C%3C%2Fp%3E%3Cp%3E%5Cbegin%7Barray%7D%7Bccc%7D%3C%2Fp%3E%3Cp%3E-3%20%26%200%20%26%2015%20%5C%5C%3C%2Fp%3E%3Cp%3E12%20%26%20-8%20%26%20-8%20%5C%5C%3C%2Fp%3E%3Cp%3E-4%20%26%20-8%20%26%202%3C%2Fp%3E%3Cp%3E%5Cend%7Barray%7D%3C%2Fp%3E%3Cp%3E%5Cright%7C%3D%5C%5C%3D-3%2A%5Cleft%7C%3C%2Fp%3E%3Cp%3E%5Cbegin%7Barray%7D%7Bcc%7D%3C%2Fp%3E%3Cp%3E-8%20%26%20-8%20%5C%5C%3C%2Fp%3E%3Cp%3E-8%20%26%202%3C%2Fp%3E%3Cp%3E%5Cend%7Barray%7D%3C%2Fp%3E%3Cp%3E%5Cright%7C%2B15%2A%5Cleft%7C%3C%2Fp%3E%3Cp%3E%5Cbegin%7Barray%7D%7Bcc%7D%3C%2Fp%3E%3Cp%3E12%20%26%20-8%20%5C%5C%3C%2Fp%3E%3Cp%3E-4%20%26%20-8%3C%2Fp%3E%3Cp%3E%5Cend%7Barray%7D%3C%2Fp%3E%3Cp%3E%5Cright%7C%3D-3%28-16-64%29%2B15%28-96-32%29%3D240-1920%3D-1680)
г)
Координаты:
![\vec{b}=(-3,2,2)\\</p><p>\vec{c}=(-2,-4,1)](https://tex.z-dn.net/?f=%5Cvec%7Bb%7D%3D%28-3%2C2%2C2%29%5C%5C%3C%2Fp%3E%3Cp%3E%5Cvec%7Bc%7D%3D%28-2%2C-4%2C1%29)
Векторы коллинеарны, если их соответствующие кординаты пропорциональны
Проверим это утверждение:
![\frac{-3}{-2}\neq \frac{2}{-4}](https://tex.z-dn.net/?f=%5Cfrac%7B-3%7D%7B-2%7D%5Cneq%20%5Cfrac%7B2%7D%7B-4%7D)
Данное равенство неверно, значит векторы b и c не коллинеарны
Векторы ортогональны, если их скалярное произведение равно нулю.
Проверим это утверждение:
![\vec{b}*\vec{c}=6-8+2=0](https://tex.z-dn.net/?f=%5Cvec%7Bb%7D%2A%5Cvec%7Bc%7D%3D6-8%2B2%3D0)
- верно, значит данные векторы ортогональны
Векторы b и c ортогональны
д)
Координаты:
![7*\vec{a}=(-7,0,35)\\</p><p>2*\vec{b}=(-6,4,4)\\</p><p>(-3)*\vec{c}=(6,12,-3)](https://tex.z-dn.net/?f=7%2A%5Cvec%7Ba%7D%3D%28-7%2C0%2C35%29%5C%5C%3C%2Fp%3E%3Cp%3E2%2A%5Cvec%7Bb%7D%3D%28-6%2C4%2C4%29%5C%5C%3C%2Fp%3E%3Cp%3E%28-3%29%2A%5Cvec%7Bc%7D%3D%286%2C12%2C-3%29)
Три вектора компланарны, если их смешанное произведение равно нулю.
![(7*\vec{a},2*\vec{b},(-3)*\vec{c})=\left|</p><p>\begin{array}{ccc}</p><p>-7 & 0 & 35 \\</p><p>-6 & 4 & 4 \\</p><p>6 & 12 & -3</p><p>\end{array}</p><p>\right|=-7*\left|</p><p>\begin{array}{cc}</p><p>4 & 4 \\</p><p>12 & -3</p><p>\end{array}</p><p>\right|+35*\left|</p><p>\begin{array}{cc}</p><p>-6 & 4 \\</p><p>6 & 12</p><p>\end{array}</p><p>\right|=-7(-12-48)+35*(-72-24)=420-3360=-2940](https://tex.z-dn.net/?f=%287%2A%5Cvec%7Ba%7D%2C2%2A%5Cvec%7Bb%7D%2C%28-3%29%2A%5Cvec%7Bc%7D%29%3D%5Cleft%7C%3C%2Fp%3E%3Cp%3E%5Cbegin%7Barray%7D%7Bccc%7D%3C%2Fp%3E%3Cp%3E-7%20%26%200%20%26%2035%20%5C%5C%3C%2Fp%3E%3Cp%3E-6%20%26%204%20%26%204%20%5C%5C%3C%2Fp%3E%3Cp%3E6%20%26%2012%20%26%20-3%3C%2Fp%3E%3Cp%3E%5Cend%7Barray%7D%3C%2Fp%3E%3Cp%3E%5Cright%7C%3D-7%2A%5Cleft%7C%3C%2Fp%3E%3Cp%3E%5Cbegin%7Barray%7D%7Bcc%7D%3C%2Fp%3E%3Cp%3E4%20%26%204%20%5C%5C%3C%2Fp%3E%3Cp%3E12%20%26%20-3%3C%2Fp%3E%3Cp%3E%5Cend%7Barray%7D%3C%2Fp%3E%3Cp%3E%5Cright%7C%2B35%2A%5Cleft%7C%3C%2Fp%3E%3Cp%3E%5Cbegin%7Barray%7D%7Bcc%7D%3C%2Fp%3E%3Cp%3E-6%20%26%204%20%5C%5C%3C%2Fp%3E%3Cp%3E6%20%26%2012%3C%2Fp%3E%3Cp%3E%5Cend%7Barray%7D%3C%2Fp%3E%3Cp%3E%5Cright%7C%3D-7%28-12-48%29%2B35%2A%28-72-24%29%3D420-3360%3D-2940)
-2940 не равно нулю => данные векторы не компланарны.