Diketahui \(\overrightarrow{u}=\left( \begin{matrix}2\\6\\-3\\\end{matrix} \right)\) dan \(\overrightarrow{v}=\left( \begin{matrix}-4\\3\\0\\\end{matrix} \right)\).
(A) Tentukan proyeksi skalar orthogonal \(\overrightarrow{u}\) pada \(\overrightarrow{v}\).
(B) Tentukan proyeksi vektor orthogonal \(\overrightarrow{u}\) pada \(\overrightarrow{v}\).
(A) Tentukan proyeksi skalar orthogonal \(\overrightarrow{u}\) pada \(\overrightarrow{v}\).
\begin{equation*}
\begin{split}
\overrightarrow{|p|} & =\frac{\overrightarrow{u}\cdot \overrightarrow{v}}{\overrightarrow{|v|}} \\\\
\overrightarrow{|p|} & = \frac{\left( \begin{matrix}2\\6\\-3\\\end{matrix} \right)\cdot \left( \begin{matrix}-4\\3\\0\\\end{matrix} \right)}{\sqrt{(-4)^2 + 3^2 + 0^2}} \\\\
\overrightarrow{|p|} & =\frac{(2)(-4) + (6)(3) + (-3)(0)}{\sqrt{25}} \\\\
\overrightarrow{|p|} & =\frac{-8+18+0}{5} \\\\
\overrightarrow{|p|} & = 2
\end{split}
\end{equation*}
(B) Tentukan proyeksi vektor orthogonal \(\overrightarrow{u}\) pada \(\overrightarrow{v}\).
\begin{equation*}
\begin{split}
\overrightarrow{p} & =\frac{\overrightarrow{u} \cdot \overrightarrow{v}}{| \: \overrightarrow{v}\: |^2} \:.\: \overrightarrow{v} \\\\
\overrightarrow{p} & = \frac{\left( \begin{matrix}2\\6\\-3\\\end{matrix} \right)\cdot \left( \begin{matrix}-4\\3\\0\\\end{matrix} \right)}{(\sqrt{(-4)^2 + 3^2 + 0^2})^2} \:.\: \left( \begin{matrix}-4\\3\\0\\\end{matrix} \right)\\\\
\overrightarrow{p} & =\frac{(2)(-4) + (6)(3) + (-3)(0)}{25} \:.\: \left( \begin{matrix}-4\\3\\0\\\end{matrix} \right) \\\\
\overrightarrow{p} & =\frac{-8+18+0}{25} \:.\: \left( \begin{matrix}-4\\3\\0\\\end{matrix} \right) \\\\
\overrightarrow{p} & = \frac 25 \:.\: \left( \begin{matrix}-4\\3\\0\\\end{matrix} \right) \\\\
\end{split}
\end{equation*}
