Question

Найдите | a+b | , если |a|=5 |b|=7 и |a-b|= $sqrt{84}$

Answer 1

$(a-b)^2=a^2-2ab+b^2=|a|^2-2|a||b|cos(a, b)+|b|^2=|a-b|^2$
$5^2-2*5*7*cos (a, b)+7^2=(sqrt{84})^2$
$25-70cos(a,b)+49=84$
$cos(ab)=frac{84-25-49}{70}=frac{10}{-70}=-frac{1}{7}$

$|a+b|=sqrt{|a+b|^2}=sqrt{(a+b)^2}=sqrt{a^2+2ab+b^2}=$
$=sqrt{|a|^2+2|a||b|cos(a,b)+|b|^2}=sqrt{5^2+2*5*7*(-frac{1}{7})+7^2}=$
$=sqrt{25-10+49}=sqrt{64}=8$
ответ: 8