Proyeksi Vektor

Misal hasil vektor proyeksi \(\textbf{u}\) pada \(\textbf{v}\) adalah \(\textbf{w}\) \(\textbf{w} = \dfrac{\textbf{u} \cdot \textbf{v}}{(||\textbf{v}||)^2} \cdot \textbf{v}\) \(\textbf{w} = \dfrac{\left(\begin{array}{c}-1\\ 2\end{array}\right) \cdot \left(\begin{array}{c}3\\ -1\end{array}\right)}{(\sqrt{3^2 + (-1)^2})^2} \cdot \left(\begin{array}{c}3\\ -1\end{array}\right)\) \(\textbf{w} = \dfrac{(-1)(3) + (2)(-1)}{3^2 + (-1)^2} \cdot \left(\begin{array}{c}3\\ -1\end{array}\right)\) \(\textbf{w} = \dfrac{-5}{10} \cdot \left(\begin{array}{c}3\\ -1\end{array}\right)\) \(\textbf{w} = -\dfrac{1}{2} \cdot \left(\begin{array}{c}3\\ -1\end{array}\right)\) \(\textbf{w} =  \left(\begin{array}{c}-\dfrac{3}{2}\\ \dfrac{1}{2}\end{array}\right)\) \(\textbf{w} = -\dfrac{3}{2}\textbf{i}+ \dfrac{1}{2}\textbf{j} \)   Jadi hasil vektor proyeksi \(\textbf{u}\) pada \(\textbf{v}\) adalah \(\textbf{w} = -\frac{3}{2}\textbf{i} + \frac{1}{2}\textbf{j} \)

Proyeksi skalar (panjang proyeksi) vektor \(\textbf{v}\) pada \(\textbf{u}\) adalah \(||\textbf{w}||\) \(||\textbf{w}|| = \dfrac{|\textbf{v} \cdot \textbf{u}|}{||\textbf{u}||}\) \(||\textbf{w}|| = \dfrac{\left|\left(\begin{array}{c}4\\ 1\end{array}\right) \cdot \left(\begin{array}{c}-2\\ -3\end{array}\right)\right|}{\sqrt{(-2)^2 + (-3)^2}}\) \(||\textbf{w}|| = \dfrac{|(4)(-2) + (1)(-3)|}{\sqrt{4 + 9}}\) \(||\textbf{w}|| = \dfrac{11}{\sqrt{13}}\times \color{red} \dfrac{\sqrt{13}}{\sqrt{13}}\) \(||\textbf{w}|| = \dfrac{11}{13}\sqrt{13}\)   Jadi proyeksi skalar \(\textbf{v}\) pada \(\textbf{u}\) adalah \(||\textbf{w}|| = \dfrac{11}{13}\sqrt{13}\) satuan