Задача 61562 ...

6192282b81a3d5111ee120c3

15.11.2021 12:30:06

Векторы а и b образуют угол φ= π/3. Зная, что |a| = 1, |b| = 2, вычислить а) c * d; б) |c x d|. с=2а+3b; d=-3a+5b

5f3ea7e3faf909182968ddd9

15.11.2021 14:52:24

★

a)
vector{c}*vector{d}=(2 vector{a}+3 vector{b})*(-3 vector{a}+5 vector{b})=

=2 vector{a}*(-3) vector{a}+3 vector{b}*(-3) vector{a}+2 vector{a}*5 vector{b}+3* vector{b}*5 vector{b}=

=-6 vector{a}* vector{a}-9 vector{b}* vector{a}+10 vector{a}* vector{b}+15 vector{b}*vector{b}=

по свойству скалярного произведения векторов: [r] vector{b}* vector{a}=vector{a}* vector{b} [/r]

=-6 vector{a}* vector{a}+ vector{a}* vector{b}+15 vector{b}*vector{b}=


=-6|vector{a}|*| vector{a}|*cos0+ |vector{a}|* |vector{b}|*cos(π/3)+15 |vector{b}|*|vector{b}}*cos0=

=-6*1*1*1+1*2*(1/2)+15*2*2*1=-6+1+60=[b]55[/b]


б)
[vector{c} × vector{d}]=[(2* vector{a}+3* vector{b}) × (-3* vector{a}+5* vector{b})]=

=[2* vector{a} × (-3)* vector{a}]+[3* vector{b} × (-3)* vector{a}]+[2 *vector{a} × 5* vector{b}]+[3* vector{b} × 5 *vector{b}]=

=-6 [vector{a} × vector{a}]-9 [vector{b} × vector{a}]+10[ vector{a} × vector{b}]+15[ vector{b} × vector{b}]=

по свойству векторного произведения векторов: [r] [vector{b} × vector{a}]=-[ vector{a} × vector{b}][/r]


=-6 [vector{a} × vector{a}]-9 (-[vector{a} × vector{b}]+10[ vector{a} × vector{b}]+15[ vector{b} × vector{b}]=


=-6 [vector{a} × vector{a}]+19[vector{a} × vector{b}]+15[ vector{b} × vector{b}]



|[vector{c} × vector{d}]|=|-6|*|vector{a}|*| vector{a}|*sin0+ |vector{a}|* |vector{b}|*sin(π/3)+15 |vector{b}|*|vector{b}}*sin0=

=6*1*1*0+19*1*2*(sqrt(3)/2)+15*2*2*0=[b]19sqrt(3)[/b]