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A B C a
a) \(\overrightarrow{AB}.\overrightarrow{AC}=AB.AC.cos\left(\overrightarrow{AB},\overrightarrow{AC}\right)=a.a.cos60^o=a.a.\dfrac{1}{2}\)\(=\dfrac{a^2}{2}\).
\(\overrightarrow{AB}.\overrightarrow{BC}=-\overrightarrow{BA}.\overrightarrow{BC}==-a.a.cos\left(\overrightarrow{BA},\overrightarrow{BC}\right)\)\(=-a.a.cos60^o=-\dfrac{a^2}{2}\).
\(\overrightarrow{AB}.\overrightarrow{AC}=\left|\overrightarrow{AB}\right|.\left|\overrightarrow{AC}\right|.cos\left(\overrightarrow{AB},\overrightarrow{AC}\right)=a.a.cos60=\dfrac{1}{2}a^2\)\(\overrightarrow{AB}.\overrightarrow{BC}=-\overrightarrow{BA}.\overrightarrow{BC}=-\left(\overrightarrow{BA}.\overrightarrow{BC}\right)=-\left(\left|\overrightarrow{BA}\right|.\left|\overrightarrow{BC}\right|.cos\left(\overrightarrow{BA},\overrightarrow{BC}\right)\right)=-\left(a.a.cos60\right)=-\dfrac{1}{2}a^2\)
A B C
a) \(\overrightarrow{AB}.\overrightarrow{AC}=0\) do \(AB\perp AC\).
b)
\(BC=\sqrt{AB^2+AC^2}=\sqrt{a^2+a^2}=\sqrt{2}a\).
\(\overrightarrow{BA}.\overrightarrow{BC}=BA.BC.cos\left(\overrightarrow{BA},\overrightarrow{BC}\right)=a.\sqrt{2}a.cos45^o=a^2\).
c) \(\overrightarrow{AB}.\overrightarrow{BC}=-\overrightarrow{BA}.\overrightarrow{BC}=-a^2\).
a) Có \(\overrightarrow{BC}^2=\left(\overrightarrow{AC}-\overrightarrow{AB}\right)^2=\overrightarrow{AC}^2+\overrightarrow{AB}^2-2\overrightarrow{AC}.\overrightarrow{AB}\)
Suy ra: \(\overrightarrow{AC}.\overrightarrow{AB}=\dfrac{\overrightarrow{AC^2}+\overrightarrow{AB}^2-\overrightarrow{BC}^2}{2}=\dfrac{8^2+6^2-11^2}{2}=-\dfrac{21}{2}\).
Do \(\overrightarrow{AC}.\overrightarrow{AB}< 0\) nên \(cos\widehat{BAC}< 0\) suy ra góc A là góc tù.
b) Từ câu a suy ra: \(cos\widehat{BAC}=\dfrac{\overrightarrow{AB}.\overrightarrow{AC}}{\left|\overrightarrow{AB}\right|.\left|\overrightarrow{AC}\right|}=-\dfrac{21}{2.6.8}=-\dfrac{7}{32}\).
Do N là trung điểm của AC nên \(AN=AC:2=8:2=4cm\).
\(\overrightarrow{AM}.\overrightarrow{AN}=AM.AN.cos\left(\overrightarrow{AM},\overrightarrow{AN}\right)\)
\(=2.4.cos\left(\overrightarrow{AB},\overrightarrow{AC}\right)=2.4.\dfrac{-7}{32}=-\dfrac{7}{4}\).
AB.AC= |AB||AC|.cos(ab,ac)=-35/2
AB.BC=AB(BA+AC)=-49+ -35/2=-133/2
