Diketahui koordinat titik A \((3, -1, 2)\), B \((-7, 4, -5)\) dan C \((2, 6, -15)\).
Tentukan nilai dari \(\overrightarrow{AB}\cdot \overrightarrow{BC}\)
Vektor \(\overrightarrow{AB}\)
\begin{equation*} \begin{split} \overrightarrow{AB} & = \overrightarrow{b}-\overrightarrow{a} \\\\ \overrightarrow{AB} & = \left(\begin{matrix}-7\\4\\-5\\ \end{matrix}\right)-\left(\begin{matrix}3\\-1\\2\\ \end{matrix}\right) \\\\ \overrightarrow{AB} & =\left(\begin{matrix}-10\\5\\-7\\ \end{matrix}\right) \end{split} \end{equation*}
Vektor \(\overrightarrow{BC}\)
\begin{equation*} \begin{split} \overrightarrow{BC} & =\overrightarrow{c}-\overrightarrow{b} \\\\ \overrightarrow{BC} & =\left(\begin{matrix}2\\6\\-15\\\end{matrix}\right)-\left(\begin{matrix}-7\\4\\-5\\\end{matrix}\right) \\\\ \overrightarrow{BC} & =\left(\begin{matrix}9\\2\\-10\\\end{matrix}\right) \end{split} \end{equation*}
\begin{equation*}
\begin{split}
\overrightarrow{AB} \:.\: \overrightarrow{BC} & = \left(\begin{matrix}-10\\5\\-7\\ \end{matrix}\right) \cdot \left(\begin{matrix}9\\2\\-10\\\end{matrix}\right) \\\\
\overrightarrow{AB} \:.\: \overrightarrow{BC} & = –10 \:.\: 9 + 5 \:.\: 2 + –7 \:.\: –10 \\\\
\overrightarrow{AB} \:.\: \overrightarrow{BC} & = –90 + 10 + 70 \\\\
\overrightarrow{AB} \:.\: \overrightarrow{BC} & = –10
\end{split}
\end{equation*}