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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: \(\overrightarrow a .\overrightarrow b = 1.3 + 2\left( { – 4} \right) = 3 – 8 = – 5\)
\(\overrightarrow b .\overrightarrow c = 3\left( { – 5} \right) + \left( { – 4} \right).3 = – 15 – 12 = – 27\)
\(\overrightarrow c .\overrightarrow a = – 5.1 + 3.2 = – 5 + 6 = 1\)
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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: \(\overrightarrow a .\left( {\overrightarrow b + \overrightarrow c } \right) = 1.\left( { – 2} \right) + 2\left( { – 1} \right) = – 2 – 2 = – 4\)
\( \Rightarrow \) \(\cos \left( {\overrightarrow a ;\,\,\overrightarrow b + \overrightarrow c } \right) = \frac{{\overrightarrow a .\left( {\overrightarrow b + \overrightarrow c } \right)}}{{\left| {\overrightarrow a } \right|.\left| {\overrightarrow b + \overrightarrow c } \right|}} = \frac{{ – 4}}{{\sqrt 5 .\sqrt 5 }} = \frac{{ – 4}}{5}\)
\( \Rightarrow \) \(\left( {\overrightarrow a ;\,\,\overrightarrow b + \overrightarrow c } \right) \approx {143^ \circ }\)