1. Векторы коллинеарны тогда, когда их соответствующие координаты пропорциональны
![\dfrac{-2}{1}=\dfrac{8}{-4}=\dfrac{-4}{k}~~\Leftrightarrow~~~ k=2](https://tex.z-dn.net/?f=%5Cdfrac%7B-2%7D%7B1%7D%3D%5Cdfrac%7B8%7D%7B-4%7D%3D%5Cdfrac%7B-4%7D%7Bk%7D~~%5CLeftrightarrow~~~+k%3D2)
Векторы перпендикулярны тогда, когда скалярное произведение векторов равно нулю.
![(-2)\cdot 1+8\cdot(-4)+(-4)\cdot k=0\\ -2-32-4k=0\\ k=-8.5](https://tex.z-dn.net/?f=%28-2%29%5Ccdot+1%2B8%5Ccdot%28-4%29%2B%28-4%29%5Ccdot+k%3D0%5C%5C+-2-32-4k%3D0%5C%5C+k%3D-8.5)
2. a) ![\overline{AB}=\{3-(-2);-2-1;-1-3\}=\{5;-3;-4\}](https://tex.z-dn.net/?f=%5Coverline%7BAB%7D%3D%5C%7B3-%28-2%29%3B-2-1%3B-1-3%5C%7D%3D%5C%7B5%3B-3%3B-4%5C%7D)
![\overline{AC}=\{-1-(-2);4-1;2-3\}=\{1;3;-1\}](https://tex.z-dn.net/?f=%5Coverline%7BAC%7D%3D%5C%7B-1-%28-2%29%3B4-1%3B2-3%5C%7D%3D%5C%7B1%3B3%3B-1%5C%7D)
б) ![|\overline{AB}|=\sqrt{5^2+(-3)^2+(-4)^2}=5\sqrt{2}](https://tex.z-dn.net/?f=%7C%5Coverline%7BAB%7D%7C%3D%5Csqrt%7B5%5E2%2B%28-3%29%5E2%2B%28-4%29%5E2%7D%3D5%5Csqrt%7B2%7D)
в) ![\overline{MN}=2\overline{AB}-3\overline{AC}=\{2\cdot 5-3\cdot1; 2\cdot(-3)-3\cdot3;2\cdot(-4)-3\cdot(-1)\}=\\ =\{10-3;-6-9;-8+3\}=\{7;-15;-5\}](https://tex.z-dn.net/?f=%5Coverline%7BMN%7D%3D2%5Coverline%7BAB%7D-3%5Coverline%7BAC%7D%3D%5C%7B2%5Ccdot+5-3%5Ccdot1%3B+2%5Ccdot%28-3%29-3%5Ccdot3%3B2%5Ccdot%28-4%29-3%5Ccdot%28-1%29%5C%7D%3D%5C%5C+%3D%5C%7B10-3%3B-6-9%3B-8%2B3%5C%7D%3D%5C%7B7%3B-15%3B-5%5C%7D)
3. ![\overline{n}=3\overline{a}-2\overline{b}=\{2\cdot 3-2\cdot3;3\cdot3-2\cdot(-1);3\cdot(-1)-2\cdot 0\}=\{0;11; -3\}](https://tex.z-dn.net/?f=%5Coverline%7Bn%7D%3D3%5Coverline%7Ba%7D-2%5Coverline%7Bb%7D%3D%5C%7B2%5Ccdot+3-2%5Ccdot3%3B3%5Ccdot3-2%5Ccdot%28-1%29%3B3%5Ccdot%28-1%29-2%5Ccdot+0%5C%7D%3D%5C%7B0%3B11%3B+-3%5C%7D)
4. Найдем векторы АВ и АС и потом уже найдем угол между векторами AB,AC.
![\overline{AB}=\{-2-1;4-0;2-2\}=\{-3;4;0\}\\ \overline{AC}=\{3-1;1-0;0-2\}=\{2;1;-2\}](https://tex.z-dn.net/?f=%5Coverline%7BAB%7D%3D%5C%7B-2-1%3B4-0%3B2-2%5C%7D%3D%5C%7B-3%3B4%3B0%5C%7D%5C%5C+%5Coverline%7BAC%7D%3D%5C%7B3-1%3B1-0%3B0-2%5C%7D%3D%5C%7B2%3B1%3B-2%5C%7D)
![\cos\angle(\overline{AB},\overline{AC})=\dfrac{\overline{AB}\cdot \overline{AC}}{|\overline{AB}|\cdot |\overline{AC}|}=\dfrac{(-3)\cdot2+4\cdot1+0\cdot(-2)}{\sqrt{(-3)^2+4^2+0^2}\cdot\sqrt{2^2+1^2+(-2)^2}}=-\dfrac{2}{15}](https://tex.z-dn.net/?f=%5Ccos%5Cangle%28%5Coverline%7BAB%7D%2C%5Coverline%7BAC%7D%29%3D%5Cdfrac%7B%5Coverline%7BAB%7D%5Ccdot+%5Coverline%7BAC%7D%7D%7B%7C%5Coverline%7BAB%7D%7C%5Ccdot+%7C%5Coverline%7BAC%7D%7C%7D%3D%5Cdfrac%7B%28-3%29%5Ccdot2%2B4%5Ccdot1%2B0%5Ccdot%28-2%29%7D%7B%5Csqrt%7B%28-3%29%5E2%2B4%5E2%2B0%5E2%7D%5Ccdot%5Csqrt%7B2%5E2%2B1%5E2%2B%28-2%29%5E2%7D%7D%3D-%5Cdfrac%7B2%7D%7B15%7D)
1) {х²+у²=58
{ху=21
х²+2ху+у²-2ху=58
(х+у)²-2×21=58
(х+у)²=58+42
(х+у)²=100
{(х+у)=10
{ху=21
х=(10-у)
ху=21
(10-у)×у=21
10у-у²=21
у²-10у+21=0
По теореме Виета:
у1+у2=-(-10)=10
у1×у2=21
у1=3
у2=7
х1=10-у1
х1=10-3
х1=7
х2=10-у2
х2=10-7
х2=3
(7;3) и (3;7)
Проверка:
х1²+у1²=58
7²+3²=58
49+9=58
58=58-истина
х1×у1=21
7×3=21
21=21-истина.
х2²+у2²=58
3²+7²=58
9+49=58
58=58-истина
х2×у2=21
3×7=21
21=21-истина.
2) {x²+y²= 41
{xy = 20
х²+2ху+у²-2ху=41
(х+у)²-2ху=41
(х+у)²=41+2ху
(х+у)²=41+2×20=41+40=81
(х+у)²=81
{х+у=9
{ху=20
х=(9-у)
у×(9-у)=20
9у-у²=20
у²-9у+20=0
По теореме Виета:
у1+у2=-(-9)=9
у1×у2=20
у1=4
у2=5
х1=9-у1
х1=9-4
х1=5
х2=9-у2
х2=9-5
х2=4
(5;4) и (4;5)
Проверка:
х1²+у1²=41
5²+4²=41
25+16=41
41=41-истина.
х1×у1=20
5×4=20
20=20-истина.
х2²+у2²=41
4²+5²=41
16+25=41
41=41-истина
х2×у2=20
4×5=20
20=20-истина.
= sin^2 t - cos^2 t /(- ctg t * tg t) = sin^2 t - cos^2 t / - 1 =
= sin^2 t + cos^2 t = 1
Все,что могу
Прости,что так мало,но это хотя бы на 3
Удачи:)