1) 75/100 * 14/15 = 7/10
2)7/10 : 12/10 = 7/10 * 10/12 = 7/12
3) 7/8:7/12 = 7/8 * 12/7 = 12/8 = 3/2 = 1,5
1. 684:76= 9(кол-во)
2. 49*9= 441
3. 684+441=1125
5a+2a=35
7a=35
a=5
b*b+9=34
b²=25
b=5;b=-5
1.
![\overrightarrow {AB} = \overrightarrow {(3-0; -1-2; 5-1)} = \overrightarrow {(3; -3; 4)}](https://tex.z-dn.net/?f=%5Coverrightarrow+%7BAB%7D+%3D+%5Coverrightarrow+%7B%283-0%3B+-1-2%3B+5-1%29%7D+%3D+%5Coverrightarrow+%7B%283%3B+-3%3B+4%29%7D)
2.
![\overrightarrow {A_1C_1} +\overrightarrow {D_1D}+\overrightarrow {CD}= \overrightarrow {A_1C_1} +\overrightarrow {C_1D_1}+ \overrightarrow {D_1D}}= \overrightarrow {A_1D}](https://tex.z-dn.net/?f=%5Coverrightarrow+%7BA_1C_1%7D+%2B%5Coverrightarrow+%7BD_1D%7D%2B%5Coverrightarrow+%7BCD%7D%3D+%5Coverrightarrow+%7BA_1C_1%7D+%2B%5Coverrightarrow+%7BC_1D_1%7D%2B+%5Coverrightarrow+%7BD_1D%7D%7D%3D+%5Coverrightarrow+%7BA_1D%7D)
4. Не понял по каким векторам... Вроде должно быть по вершинам.
5.
![\frac{1}{b} = (\frac {1}{1}; \frac {1}{-1}; \frac {1}{4}) = (1; -1; \frac {1}{4}) \\ \frac{1}{b} -a = (1-5; -1+2; \frac {1}{4}-3)=(-4; 1; -\frac{11}{4}) \\ 3a \cdot ( \frac{1}{b} -a) =5 \cdot (-4)+(-2) \cdot 1+ 3 \cdot (-\frac{11}{4})= -20-2-\frac{33}{4} = -30 \frac {1}{4}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7Bb%7D+%3D+%28%5Cfrac+%7B1%7D%7B1%7D%3B+%5Cfrac+%7B1%7D%7B-1%7D%3B+%5Cfrac+%7B1%7D%7B4%7D%29+%3D+%281%3B+-1%3B+%5Cfrac+%7B1%7D%7B4%7D%29+%5C%5C%0A%5Cfrac%7B1%7D%7Bb%7D+-a+%3D+%281-5%3B+-1%2B2%3B+%5Cfrac+%7B1%7D%7B4%7D-3%29%3D%28-4%3B+1%3B+-%5Cfrac%7B11%7D%7B4%7D%29+%5C%5C%0A3a+%5Ccdot+%28+%5Cfrac%7B1%7D%7Bb%7D+-a%29++%3D5+%5Ccdot+%28-4%29%2B%28-2%29+%5Ccdot+1%2B+3+%5Ccdot+%28-%5Cfrac%7B11%7D%7B4%7D%29%3D+-20-2-%5Cfrac%7B33%7D%7B4%7D+%3D+-30+%5Cfrac+%7B1%7D%7B4%7D)
6. Коллинеарны когда:
![\lambda = \frac{-1}{-4}=\frac{1}{4} \\ 2m= \frac{1}{\frac{1}{4}}=4 \\ m=2](https://tex.z-dn.net/?f=%5Clambda+%3D+%5Cfrac%7B-1%7D%7B-4%7D%3D%5Cfrac%7B1%7D%7B4%7D+%5C%5C%0A2m%3D+%5Cfrac%7B1%7D%7B%5Cfrac%7B1%7D%7B4%7D%7D%3D4+%5C%5C%0Am%3D2)
Перпендикулярны если скалярное произведение равно нулю:
![\langle a,b \rangle =2m \cdot 1+(-4) \cdot (-1)+(-3) \cdot 1 = 2m+4-3=2m-1=0 \\ m= \frac {1}{2}](https://tex.z-dn.net/?f=%5Clangle+a%2Cb+%5Crangle+%3D2m+%5Ccdot+1%2B%28-4%29+%5Ccdot+%28-1%29%2B%28-3%29+%5Ccdot+1+%3D+2m%2B4-3%3D2m-1%3D0+%5C%5C%0Am%3D+%5Cfrac+%7B1%7D%7B2%7D)
7.
![cos \alpha = \frac {\langle a,b \rangle}{L_a \cdot L_b} = \frac {3 \cdot 5 + 0 \cdot 0 + (-4) \cdot (-12)}{\sqrt{3^2+0^2+(-4)^2} \cdot \sqrt{5^2+0^2+(-12)^2}}= \frac {15+48}{\sqrt {25} \cdot \sqrt{169}}= \frac{63}{5 \cdot 13} = \frac {63}{65}](https://tex.z-dn.net/?f=cos+%5Calpha++%3D+%5Cfrac+%7B%5Clangle+a%2Cb+%5Crangle%7D%7BL_a+%5Ccdot+L_b%7D+%3D+%5Cfrac+%7B3+%5Ccdot+5+%2B+0+%5Ccdot+0+%2B+%28-4%29+%5Ccdot+%28-12%29%7D%7B%5Csqrt%7B3%5E2%2B0%5E2%2B%28-4%29%5E2%7D+%5Ccdot+%5Csqrt%7B5%5E2%2B0%5E2%2B%28-12%29%5E2%7D%7D%3D+%5Cfrac+%7B15%2B48%7D%7B%5Csqrt+%7B25%7D+%5Ccdot+%5Csqrt%7B169%7D%7D%3D+%5Cfrac%7B63%7D%7B5+%5Ccdot+13%7D+%3D+%5Cfrac+%7B63%7D%7B65%7D)
1)(7/5-1/6)+2/5=(42/30-5/30)+2/5=38/30+2/5=38/30+12/30=50/30=1 20/30=1 1/3
2)
(3/8-1/9)+35/36=(27/72-8/72)+35/36=19/72+70/72=89/72=1 17/72
3)(8/9-5/6)+2/3=(16/18-15/18)+2/3=1/18+2/3=1/18+12/18=13/18
4)
(7/8-13/20)+9/10=(35/40-26/40)+9/10=9/40+9/10=9/40+36/40=45/40=1 5/40=1 1:8
5)
(7/18-1/12)+5/6=(14/36-3/36)+5/6=11/36+5/6=11/36+30/36=41/36=1 5/36
6)
(3/4-8/15)+17/20=(45/60-32/60)+17/20=13/60+17/20=13/60+51/60=64/60=1 4/60=1 1:15