Question

cho 2 vecto \(\overrightarrow{a},\overrightarrow{b}\) thoa man \(\left|\overrightarrow{a}\right|=4,\left|\overrightarrow{b}\right|=3\) và hai vecto \(\overrightarrow{u}=2\overrightarrow{a}+3\overrightarrow{b}\) và \(\overrightarrow{v}=-15\overrightarrow{a}+14\overrightarrow{b}\) vuông góc với nhau. Tính \(\left(\overrightarrow{a},\overrightarrow{b}\right)=???\)

Answer 1

\(u.v=0\Leftrightarrow\left(2a+3b\right)\left(-15a+14b\right)=0\)

\(\Leftrightarrow-30a^2+42b^2-17ab=0\)

\(\Leftrightarrow ab=\frac{-30.4^2+42.3^2}{17}=-6\)

\(\Rightarrow cos\left(a;b\right)=\frac{ab}{\left|a\right|\left|b\right|}=-\frac{6}{12}=-\frac{1}{2}\Rightarrow\left(a;b\right)=120^0\)