a) Ta có:

\(\overrightarrow{AM}=\overrightarrow{AB}+\overrightarrow{BM}\)

         \(=\overrightarrow{AB}+k\overrightarrow{BC}\)

         \(=\overrightarrow{AB}+k\left(\overrightarrow{AC}-\overrightarrow{AB}\right)\)

         \(=\left(1-k\right)\overrightarrow{AB}+k\overrightarrow{AC}\)

b) \(\overrightarrow{NP}=\overrightarrow{AP}-\overrightarrow{AN}\)

             \(=\dfrac{2}{3}\overrightarrow{AC}-\dfrac{3}{4}\overrightarrow{AB}\)

Để \(AM\perp NP\)

\(\Rightarrow\overrightarrow{AM}.\overrightarrow{NP}=\overrightarrow{0}\)

\(\Rightarrow\left[\left(1-k\right)\overrightarrow{AB}+k\overrightarrow{AC}\right]\left(-\dfrac{3}{4}\overrightarrow{AB}+\dfrac{2}{3}\overrightarrow{AC}\right)=\overrightarrow{0}\)

\(\Leftrightarrow\dfrac{3\left(k-1\right)}{4}AB^2+\dfrac{2k}{3}AC^2+\dfrac{2\left(1-k\right)}{3}\overrightarrow{AB}.\overrightarrow{AC}-\dfrac{3k}{4}\overrightarrow{AB}.\overrightarrow{AC}=\overrightarrow{0}\)

\(\Leftrightarrow\dfrac{3\left(k-1\right)}{4}AB^2+\dfrac{2k}{3}AB^2+\dfrac{1-k}{3}AB^2-\dfrac{3k}{8}AB^2=0\)

\(\Leftrightarrow AB^2\left[\dfrac{3\left(k-1\right)}{4}+\dfrac{2k}{3}+\dfrac{1-k}{3}-\dfrac{3k}{8}\right]=0\)

\(\Leftrightarrow18\left(k-1\right)+16k+8\left(1-k\right)-9k=0\left(AB>0\right)\)

\(\Leftrightarrow17k=10\)

\(\Leftrightarrow k=\dfrac{10}{17}\)

AN=2NC

=>\(AN=\frac23AC\)

Xét ΔAMN có AK là đường trung tuyến

nên \(\overrightarrow{AK}=\frac12\left(\overrightarrow{AM}+\overrightarrow{AN}\right)\)

\(=\frac12\left(\frac12\cdot\overrightarrow{AB}+\frac23\cdot\overrightarrow{AC}\right)=\frac14\cdot\overrightarrow{AB}+\frac13\cdot\overrightarrow{AC}\)

Gọi O là trung điểm của AM

BM=BC/2=a/2

\(\Leftrightarrow AM=\dfrac{a\sqrt{3}}{2}\)

\(\Leftrightarrow MO=\dfrac{a\sqrt{3}}{4}\)

Xét ΔOMB vuông tại M có 

\(BO^2=OM^2+BM^2\)

\(=a^2\cdot\dfrac{3}{16}+a^2\cdot\dfrac{1}{4}=a^2\cdot\dfrac{7}{16}\)

\(\Leftrightarrow BO=\dfrac{a\sqrt{7}}{4}\)

Xét ΔBMA có BO là đường trung tuyến

nên \(\overrightarrow{BM}+\overrightarrow{BA}=2\cdot\overrightarrow{BO}\)

\(\Leftrightarrow\left|\overrightarrow{BM}+\overrightarrow{BA}\right|=\dfrac{a\sqrt{7}}{2}\)

Đề thiếu ngay câu đầu nên ko thể giải được:

Sao cho \(?=3MB\)

3.

\(\overrightarrow{CI}=\frac{1}{2}\overrightarrow{CM}+\frac{1}{2}\overrightarrow{CN}=\frac{1}{2}.\frac{3}{4}\overrightarrow{CB}+\frac{1}{2}.\frac{1}{2}\overrightarrow{CA}=\frac{3}{8}\left(\overrightarrow{CA}+\overrightarrow{AB}\right)+\frac{1}{4}\overrightarrow{CA}\)

\(=\frac{5}{8}\overrightarrow{CA}+\frac{3}{8}\overrightarrow{AB}=\frac{3}{8}\overrightarrow{AB}-\frac{5}{8}\overrightarrow{AC}\)

Đặt \(\overrightarrow{CK}=k.\overrightarrow{CI}=\frac{3k}{8}\overrightarrow{AB}-\frac{5k}{8}\overrightarrow{AC}\)

\(\overrightarrow{BK}=\overrightarrow{BC}+\overrightarrow{CK}=\overrightarrow{BA}+\overrightarrow{AC}+\overrightarrow{CK}=-\overrightarrow{AB}+\overrightarrow{AC}+\frac{3k}{8}\overrightarrow{AB}-\frac{5k}{8}\overrightarrow{AC}\)

\(=\frac{3k-8}{8}\overrightarrow{AB}-\frac{5k-8}{8}\overrightarrow{AC}=-2\left(3k-8\right)\left(-\frac{1}{16}\overrightarrow{AB}+\frac{5k-8}{16\left(3k-8\right)}\overrightarrow{AC}\right)\)

Do B;E;K thẳng hàng nên:

\(\frac{5k-8}{16\left(3k-8\right)}=\frac{1}{3}\Rightarrow k=\frac{104}{33}\)

\(\Rightarrow\frac{KI}{KC}=\frac{71}{104}\)

Cách tính toán là như vậy, còn quá trình tính toán đúng hay sai thì bạn tự tính lại