Question

Найдите | a+b | , если |a|=5 |b|=7 и |a-b|=

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