Đáp án:
\(m=10,n=-24\)
Giải thích các bước giải:
\[\begin{array}{l}
IB = 3IC \Rightarrow \overrightarrow {IB} = 3\overrightarrow {IC} \\
\overrightarrow {IA} = \overrightarrow {IB} + \overrightarrow {BA} = \dfrac{3}{2}\overrightarrow {CB} + \overrightarrow {BA} = \dfrac{3}{2}\left( {\overrightarrow {AB} - \overrightarrow {AC} } \right) - \overrightarrow {AB} \\
= \dfrac{1}{2}\overrightarrow {AB} - \dfrac{3}{2}\overrightarrow {AC} \\
\overrightarrow {JK} = \overrightarrow {JA} + \overrightarrow {AK} = \dfrac{2}{3}\overrightarrow {CA} + \dfrac{1}{4}\overrightarrow {AB} = - \dfrac{2}{3}\overrightarrow {AC} + \dfrac{1}{4}\overrightarrow {AB} \\
\Rightarrow m\overrightarrow {IA} + n\overrightarrow {JK} = m\left( {\dfrac{1}{2}\overrightarrow {AB} - \dfrac{3}{2}\overrightarrow {AC} } \right) + n\left( { - \dfrac{2}{3}\overrightarrow {AC} + \dfrac{1}{4}\overrightarrow {AB} } \right)\\
= \left( {\dfrac{m}{2} + \dfrac{n}{4}} \right)\overrightarrow {AB} - \left( {\dfrac{{3m}}{2} + \dfrac{{2n}}{3}} \right)\overrightarrow {AC} \\
\overrightarrow {BC} = \overrightarrow {AC} - \overrightarrow {AB} = - \overrightarrow {AB} + \overrightarrow {AC} \\
\overrightarrow {BC} = m\overrightarrow {IA} + n\overrightarrow {JK} \Leftrightarrow \left\{ \begin{array}{l}
\dfrac{m}{2} + \dfrac{n}{4} = - 1\\
- \left( {\dfrac{{3m}}{2} + \dfrac{{2n}}{3}} \right) = 1
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
m = 10\\
n = - 24
\end{array} \right.
\end{array}\]
