Trong mặt phẳng tọa độ \(Oxy\) cho ba vectơ \(\overrightarrow a = (1;2),\,\,\overrightarrow b = (3; - 4),\,\,\overrightarrow c = ( - 5;3).\)
a) Tính các tích vô hướng \(\overrightarrow a .\overrightarrow b ,\,\,\overrightarrow b .\overrightarrow c ,\,\,\overrightarrow c .\overrightarrow a \)
b) Tìm góc giữa hai vectơ \(\overrightarrow a \) và \(\overrightarrow b + \overrightarrow c \)
a) Ta có: \(\overrightarrow a .\overrightarrow b = 1.3 + 2\left( { - 4} \right) = 3 - 8 = - 5\)
Advertisements (Quảng cáo)
\(\overrightarrow b .\overrightarrow c = 3\left( { - 5} \right) + \left( { - 4} \right).3 = - 15 - 12 = - 27\)
\(\overrightarrow c .\overrightarrow a = - 5.1 + 3.2 = - 5 + 6 = 1\)
b) Ta có: \(\left| {\overrightarrow a } \right| = \sqrt {{1^2} + {2^2}} = \sqrt 5 \)
\(\overrightarrow b + \overrightarrow c = \left( {3; - 4} \right) + \left( { - 5;3} \right) = \left( { - 2; - 1} \right)\) \( \Rightarrow \) \(\left| {\overrightarrow b + \overrightarrow c } \right| = \sqrt {{{\left( { - 2} \right)}^2} + {{\left( { - 1} \right)}^2}} = \sqrt 5 \)
Ta có: \(\overrightarrow a .\left( {\overrightarrow b + \overrightarrow c } \right) = 1.\left( { - 2} \right) + 2\left( { - 1} \right) = - 2 - 2 = - 4\)
\( \Rightarrow \) \(\cos \left( {\overrightarrow a ;\,\,\overrightarrow b + \overrightarrow c } \right) = \frac{{\overrightarrow a .\left( {\overrightarrow b + \overrightarrow c } \right)}}{{\left| {\overrightarrow a } \right|.\left| {\overrightarrow b + \overrightarrow c } \right|}} = \frac{{ - 4}}{{\sqrt 5 .\sqrt 5 }} = \frac{{ - 4}}{5}\)
\( \Rightarrow \) \(\left( {\overrightarrow a ;\,\,\overrightarrow b + \overrightarrow c } \right) \approx {143^ \circ }\)