If \(A,B,C\) are the vertices of a triangle \(ABC\),
\[
AB=2,\qquad BC=3,\qquad CA=4,
\]
then
\[
\overrightarrow{AB}\cdot\overrightarrow{BC}
+
\overrightarrow{BC}\cdot\overrightarrow{CA}
+
\overrightarrow{CA}\cdot\overrightarrow{AB}
=
\]
Show Hint
Whenever you see
\[
\overrightarrow{AB}\cdot\overrightarrow{BC}
+
\overrightarrow{BC}\cdot\overrightarrow{CA}
+
\overrightarrow{CA}\cdot\overrightarrow{AB},
\]
use
\[
\overrightarrow{AB}
+\overrightarrow{BC}
+\overrightarrow{CA}
=0
\]
and square both sides.