Задача 68382 Найти скалярное произведение векторов...

63c44e6903831b47663c8e27

16.01.2023 18:52:31

Найти скалярное произведение векторов p=a-2b и q=2a+b, если a=2i-5j-7k , b=5i+2j-5k

5f3ea7e3faf909182968ddd9

16.01.2023 19:50:09

★

vector{p}=vector{a}-2vector{b}=(2vector{i}-5vector{j}-7vector{k})-2*(5vector{i}+2vector{j}-5vector{k})=

=2vector{i}-5vector{j}-7vector{k}-10vector{i}-4vector{j}+10vector{k}=

=-8vector{i}-9vector{j}+3vector{k}


vector{q}=2vector{a}-vector{b}=2*(2vector{i}-5vector{j}-7vector{k})-(5vector{i}+2vector{j}-5vector{k})=

=4vector{i}-10vector{j}-14vector{k}-5vector{i}-2vector{j}+5vector{k}=

=-vector{i}-12vector{j}-9vector{k}


Скалярное произведение
vector{p}*vector{q}=(-8vector{i}-9vector{j}+3vector{k})*(-vector{i}-12vector{j}-9vector{k})


=-8vector{i}*(-vector{i}-12vector{j}*-9vector{k})-9vector{j}*(-vector{i}-12vector{j}-9vector{k})+3vector{k}*(-vector{i}-12vector{j}-9vector{k})=

=8vector{i}*(-vector{i})-8vector{i}*(-12vector{j})-8vector{i}*(-9vector{k})-9vector{j}*(-vector{i})-9vector{j}*(-12vector{j})-9vector{j}*(-9vector{k})+3vector{k}*(-vector{i})+3vector{k}*(-12vector{j})+3vector{k}*(-9vector{k})=


=8vector{i}*vector{i}+96vector{i}*vector{j}+72vector{i}*vector{k}+9vector{j}*vector{i}+108vector{j}*vector{j}*+81vector{j}*vector{k}-3vector{k}*vector{i}-36vector{k}*vector{j}-27vector{k}*vector{k}=8*1+108*1-27*1=89




2 способ

Координаты векторов:

vector{a}=(2;-5;-7)
vector{b}=(5;2;-5)

vector{p}=vector{a}-2vector{b}=(2-2*5;-5-2*2;-7-2*(-5))=(-8;-9;3)

vector{q}=2vector{a}-vector{b}=(2*2-5;2*(-5)-2;2*(-7)-(-5))=(-1;-12;-9)

Скалярное произведение векторов, заданных своими координатами равно сумме произведений одноименных координат:

vector{p}*vector{q}=-8*(-1)+(-9)*(-12)+3*(-9)=8+108-27=89