Cho \(\overrightarrow{a}=\left(3;4\right)\), \(\overrightarrow{b}\)\(=\left(7;1\right)\). Tính:
a) \(\overrightarrow{a},\overrightarrow{b}\) và hệ số đo góc \(\left(\overrightarrow{a},\overrightarrow{b}\right)\)
b) Tìm tọa độ \(\overrightarrow{c}\)biết \(\overrightarrow{a}\overrightarrow{c}=15\) và \(\overrightarrow{b}\overrightarrow{c}=10\)
a, \(\left|\overrightarrow{a}\right|=\sqrt{3^2+4^2}=5\)
\(\left|\overrightarrow{b}\right|=\sqrt{7^2+1^2}=5\sqrt{2}\)
\(cos\left(\overrightarrow{a},\overrightarrow{b}\right)=\frac{3.7+4.1}{5.5\sqrt{2}}=\frac{\sqrt{2}}{2}\) \(\Rightarrow\left(\overrightarrow{a},\overrightarrow{b}\right)=45^0\)
b, Gọi \(\overrightarrow{c}\left(x;y\right)\)
\(\overrightarrow{a}.\overrightarrow{c}=15\)
\(\Leftrightarrow3x+4y=15\)
\(\overrightarrow{b}.\overrightarrow{c}=10\)
\(\Leftrightarrow7x+y=10\)
\(\Rightarrow\left\{{}\begin{matrix}x=1\\y=3\end{matrix}\right.\)