1) A(5,6,4) , B(6,9,4) , C(2,10,10) .
Уравнение плоскости, проходящей через три точки:
![\left|\begin{array}{ccc}x-x_1&y-y_1&z-z_1\\x_2-x_1&y_2-y_1&z_2-z_1\\x_3-x_1&y_3-y_1&z_3-z_1\end{array}\right| =0\\\\\\\left|\begin{array}{ccc}x-5&y-6&z-4\\1&3&0\\-3&4&6\end{array}\right| =(x-5)\cdot 18-(y-6)\cdot 6+(z-4)\cdot 13=0\\\\\\18x-6y+13z-106=0](https://tex.z-dn.net/?f=++%5Cleft%7C%5Cbegin%7Barray%7D%7Bccc%7Dx-x_1%26y-y_1%26z-z_1%5C%5Cx_2-x_1%26y_2-y_1%26z_2-z_1%5C%5Cx_3-x_1%26y_3-y_1%26z_3-z_1%5Cend%7Barray%7D%5Cright%7C+%3D0%5C%5C%5C%5C%5C%5C%5Cleft%7C%5Cbegin%7Barray%7D%7Bccc%7Dx-5%26y-6%26z-4%5C%5C1%263%260%5C%5C-3%264%266%5Cend%7Barray%7D%5Cright%7C+%3D%28x-5%29%5Ccdot+18-%28y-6%29%5Ccdot+6%2B%28z-4%29%5Ccdot+13%3D0%5C%5C%5C%5C%5C%5C18x-6y%2B13z-106%3D0)
Расстояние от точки М(1,2,3) до плоскости найдём по формуле:
![d=\frac{|\, Ax_0+By_0+Cz_0+D\, |}{\sqrt{A^2+B^2+C^2}}\\\\d=\frac{|18\cdot 1-6\cdot 2+13\cdot 3-106|}{\sqrt{18^2+6^2+13^2}}=\frac{|-61|}{ \sqrt{529} }=\frac{61}{23} =2 \frac{15}{23}](https://tex.z-dn.net/?f=d%3D%5Cfrac%7B%7C%5C%2C+Ax_0%2BBy_0%2BCz_0%2BD%5C%2C+%7C%7D%7B%5Csqrt%7BA%5E2%2BB%5E2%2BC%5E2%7D%7D%5C%5C%5C%5Cd%3D%5Cfrac%7B%7C18%5Ccdot+1-6%5Ccdot+2%2B13%5Ccdot+3-106%7C%7D%7B%5Csqrt%7B18%5E2%2B6%5E2%2B13%5E2%7D%7D%3D%5Cfrac%7B%7C-61%7C%7D%7B+%5Csqrt%7B529%7D+%7D%3D%5Cfrac%7B61%7D%7B23%7D+%3D2+%5Cfrac%7B15%7D%7B23%7D+)
2) Векторы образуют базис, если они ЛНЗ, то есть определитель, составленный из координат этих векторов отличен от 0 .
![\vec{a}=(5,1,2)\; ,\; \; \vec{b}=(8,1,-3)\; ,\; \; \vec{c}=(-1,3,2)](https://tex.z-dn.net/?f=%5Cvec%7Ba%7D%3D%285%2C1%2C2%29%5C%3B+%2C%5C%3B+%5C%3B+%5Cvec%7Bb%7D%3D%288%2C1%2C-3%29%5C%3B+%2C%5C%3B+%5C%3B+%5Cvec%7Bc%7D%3D%28-1%2C3%2C2%29)
![\Delta = \left|\begin{array}{ccc}5&1&2\\8&1&-3\\-1&3&2\end{array}\right| =5(2+9)-(16-3)+2(24+1)=92\ne 0](https://tex.z-dn.net/?f=%5CDelta+%3D++%5Cleft%7C%5Cbegin%7Barray%7D%7Bccc%7D5%261%262%5C%5C8%261%26-3%5C%5C-1%263%262%5Cend%7Barray%7D%5Cright%7C+%3D5%282%2B9%29-%2816-3%29%2B2%2824%2B1%29%3D92%5Cne+0)
Векторы
![\vec{a}\; ,\; \vec{b}\; ,\; \vec{c}](https://tex.z-dn.net/?f=%5Cvec%7Ba%7D%5C%3B+%2C%5C%3B+%5Cvec%7Bb%7D%5C%3B+%2C%5C%3B+%5Cvec%7Bc%7D)
образуют базис. Значит, вектор
![\vec{x}](https://tex.z-dn.net/?f=%5Cvec%7Bx%7D)
можно разложить по данному базису.
Найдём координаты вектора
![\vec{x}=(7,1,9)](https://tex.z-dn.net/?f=%5Cvec%7Bx%7D%3D%287%2C1%2C9%29)
в этом базисе, используя соотношение между векторами
![\vec{x}= \alpha \cdot \vec{a}+ \beta \cdot \vec{b}+\gamma \cdot \vec{c}](https://tex.z-dn.net/?f=%5Cvec%7Bx%7D%3D+%5Calpha+%5Ccdot+%5Cvec%7Ba%7D%2B+%5Cbeta+%5Ccdot+%5Cvec%7Bb%7D%2B%5Cgamma+%5Ccdot+%5Cvec%7Bc%7D)
.
В координатной форме это соотношение будет иметь вид:
![\left\{\begin{array}{c}5 \alpha +8 \beta -\gamma=7\\ \alpha + \beta +3\gamma =1\\2 \alpha -3 \beta +2\gamma=9\end{array}\right](https://tex.z-dn.net/?f=++%5Cleft%5C%7B%5Cbegin%7Barray%7D%7Bc%7D5+%5Calpha+%2B8+%5Cbeta+-%5Cgamma%3D7%5C%5C+%5Calpha+%2B+%5Cbeta+%2B3%5Cgamma+%3D1%5C%5C2+%5Calpha+-3+%5Cbeta+%2B2%5Cgamma%3D9%5Cend%7Barray%7D%5Cright+)
Решим систему методом Гаусса.
![\left(\begin{array}{ccc}1&1&3\; \; |\; 1\\5&8&-1\; |\; 7\\2&-3&2\; \; |\; 9\end{array}\right) \sim \left(\begin{array}{ccc}1&1&3\; \; |\; 1\\0&3&-16|\, 2\\0&-5&-4\; \; |7\end{array}\right) \; \left\begin{array}{ccc}\\\cdot5\\\cdot 3\end{array}\right \oplus\sim \\\\\\ \sim \left(\begin{array}{ccc}1&1&3\; \; |\; 1\\0&3&-16\, |\; 2\\0&0&-92\; |\; 31\end{array}\right) \\\\\\-92\gamma=31\; ,\; \; \gamma=- \frac{31}{92} \\\\3 \beta =2+16\gamma =2- \frac{16\cdot31}{92}=-\frac{312}{92}\; ,\; \; \beta =-\frac{312}{92\cdot 3}=-\frac{312}{276}](https://tex.z-dn.net/?f=++%5Cleft%28%5Cbegin%7Barray%7D%7Bccc%7D1%261%263%5C%3B+%5C%3B+%7C%5C%3B+1%5C%5C5%268%26-1%5C%3B+%7C%5C%3B+7%5C%5C2%26-3%262%5C%3B+%5C%3B+%7C%5C%3B+9%5Cend%7Barray%7D%5Cright%29+%5Csim+++%5Cleft%28%5Cbegin%7Barray%7D%7Bccc%7D1%261%263%5C%3B+%5C%3B+%7C%5C%3B+1%5C%5C0%263%26-16%7C%5C%2C+2%5C%5C0%26-5%26-4%5C%3B+%5C%3B+%7C7%5Cend%7Barray%7D%5Cright%29+%5C%3B+++%5Cleft%5Cbegin%7Barray%7D%7Bccc%7D%5C%5C%5Ccdot5%5C%5C%5Ccdot+3%5Cend%7Barray%7D%5Cright+%5Coplus%5Csim+%5C%5C%5C%5C%5C%5C+%5Csim+%5Cleft%28%5Cbegin%7Barray%7D%7Bccc%7D1%261%263%5C%3B+%5C%3B+%7C%5C%3B+1%5C%5C0%263%26-16%5C%2C+%7C%5C%3B+2%5C%5C0%260%26-92%5C%3B+%7C%5C%3B+31%5Cend%7Barray%7D%5Cright%29+%5C%5C%5C%5C%5C%5C-92%5Cgamma%3D31%5C%3B+%2C%5C%3B+%5C%3B+%5Cgamma%3D-+%5Cfrac%7B31%7D%7B92%7D+%5C%5C%5C%5C3+%5Cbeta+%3D2%2B16%5Cgamma+%3D2-+%5Cfrac%7B16%5Ccdot31%7D%7B92%7D%3D-%5Cfrac%7B312%7D%7B92%7D%5C%3B+%2C%5C%3B+%5C%3B+%5Cbeta+%3D-%5Cfrac%7B312%7D%7B92%5Ccdot+3%7D%3D-%5Cfrac%7B312%7D%7B276%7D+)
![\alpha =1- \beta -3\gamma =1+\frac{312}{276}+\frac{3\cdot 31}{92}=\frac{867}{276}\\\\\\\vec{x}=\frac{867}{276}\cdot \vec{a}-\frac{312}{276}\cdot \vec{b}-\frac{31}{92} \cdot \vec{c}](https://tex.z-dn.net/?f=+%5Calpha+%3D1-+%5Cbeta+-3%5Cgamma+%3D1%2B%5Cfrac%7B312%7D%7B276%7D%2B%5Cfrac%7B3%5Ccdot+31%7D%7B92%7D%3D%5Cfrac%7B867%7D%7B276%7D%5C%5C%5C%5C%5C%5C%5Cvec%7Bx%7D%3D%5Cfrac%7B867%7D%7B276%7D%5Ccdot+%5Cvec%7Ba%7D-%5Cfrac%7B312%7D%7B276%7D%5Ccdot++%5Cvec%7Bb%7D-%5Cfrac%7B31%7D%7B92%7D+%5Ccdot+%5Cvec%7Bc%7D)