Cho vecto u = vecto a +3.vecto b vuông góc với vecto v = 7.vecto a - 5.vecto b

* Hướng dẫn giải

Đáp án B

Ta có:  

$$\begin{gathered} \left\{\begin{array}{c}\overrightarrow{u}.\overrightarrow{v}=0\\ \overrightarrow{x}.\overrightarrow{y}=0\end{array}\right.\iff \left\{\begin{array}{c}\left(\overrightarrow{a}+3\overrightarrow{b}\right).\left(7\overrightarrow{a}-5\overrightarrow{b}\right)=0\\ \left(\overrightarrow{a}-4\overrightarrow{b}\right).\left(7\overrightarrow{a}-2\overrightarrow{b}\right)=0\end{array}\right. \\ \iff \left\{\begin{array}{c}7{\left|\overrightarrow{a}\right|}^2-15{\left|\overrightarrow{b}\right|}^2=-16\overrightarrow{a}.\overrightarrow{b}\\ 7{\left|\overrightarrow{a}\right|}^2+8{\left|\overrightarrow{b}\right|}^2=30\overrightarrow{a}.\overrightarrow{b}\end{array}\right. \\ \iff \left\{\begin{array}{c}{\left|\overrightarrow{b}\right|}^2=2\overrightarrow{a}.\overrightarrow{b}\\ {\left|\overrightarrow{a}\right|}^2=2\overrightarrow{a}.\overrightarrow{b}\end{array}\right.\iff \left\{\begin{array}{c}{\left|\overrightarrow{b}\right|}^2=2\overrightarrow{a}.\overrightarrow{b}\\ \left|\overrightarrow{a}\right|=\left|\overrightarrow{b}\right|\end{array}\right. \end{gathered}$$

Từ đó, ta có:

$$\begin{gathered} \cos \left(\overrightarrow{a}.\overrightarrow{b}\right)=\frac{\overrightarrow{a}.\overrightarrow{b}}{\left|\overrightarrow{a}\right|.\left|\overrightarrow{b}\right|}=\frac{\overrightarrow{a}.\overrightarrow{b}}{{\left|\overrightarrow{b}\right|}^2}=\frac{1}{2} \\ \Rightarrow \left(\overrightarrow{a},\overrightarrow{b}\right)={60}^0 \end{gathered}$$