如图, 在 $\triangle A B C$ 中, $A D \perp A B,|\overrightarrow{A D}|=2, \overrightarrow{D C}=3 \overrightarrow{B D}$, 则 $\overrightarrow{A C} \cdot \overrightarrow{A D}$ 的值为 ( )

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$\because \overrightarrow{A C}=\overrightarrow{A D}+\overrightarrow{D C}=\overrightarrow{A D}+3 \overrightarrow{B D}=\overrightarrow{A D}+3(\overrightarrow{A D}-\overrightarrow{A B})=4 \overrightarrow{A D}-3 \overrightarrow{A B}$,

$\because A D \perp A B$, 则 $\overrightarrow{A D} \cdot \overrightarrow{A B}=0$,

所以,, 故选 D.