$\begin{array}{l} \left| {\overrightarrow {AH} } \right| = AH\\ + AB = 3k,AC = 4k\left( {k > 0} \right)\\ + Pytago \Rightarrow A{B^2} + A{C^2} = B{C^2}\\ \Rightarrow 9{k^2} + 16{k^2} = 100 \Rightarrow 25{k^2} = 100 \Rightarrow k = 2\\ \Rightarrow AB = 6cm,AC = 8cm\\ \dfrac{1}{{A{H^2}}} = \dfrac{1}{{A{B^2}}} + \dfrac{1}{{A{C^2}}} = \dfrac{1}{{{6^2}}} + \dfrac{1}{{{8^2}}} = \dfrac{{25}}{{576}}\\ \Rightarrow A{H^2} = \dfrac{{576}}{{25}} \Rightarrow AH = \dfrac{{24}}{5}\left( {cm} \right)\\ \Rightarrow \left| {\overrightarrow {AH} } \right| = AH = \dfrac{{24}}{5} \end{array}$
