91Ó°ÊÓ

Consider a vector y in \({\mathbb{R}^n}\). Then,

\({\mathop{\rm y}\nolimits} = \widehat {\mathop{\rm y}\nolimits} + {\mathop{\rm z}\nolimits} \)…. (1)

Where \(\widehat {\mathop{\rm y}\nolimits} = \alpha {\mathop{\rm z}\nolimits} \) for any scalar \(\alpha \) and z denotes any vector orthogonal to u. Consider \({\mathop{\rm z}\nolimits} = {\mathop{\rm y}\nolimits} - \alpha {\mathop{\rm u}\nolimits} \), therefore, that equation (1) is satisfied. Then \({\mathop{\rm y}\nolimits} - \widehat {\mathop{\rm y}\nolimits} \) is orthogonal to u such that if the equation (1) is satisfied with z orthogonal to u such that if \(\alpha = \frac{{{\mathop{\rm y}\nolimits} \cdot {\mathop{\rm u}\nolimits} }}{{{\mathop{\rm u}\nolimits} \cdot {\mathop{\rm u}\nolimits} }}\), and \(\widehat {\mathop{\rm y}\nolimits} = \frac{{{\mathop{\rm y}\nolimits} \cdot {\mathop{\rm u}\nolimits} }}{{{\mathop{\rm u}\nolimits} \cdot {\mathop{\rm u}\nolimits} }}{\mathop{\rm u}\nolimits} \).

The vector \(\widehat {\mathop{\rm y}\nolimits} \) is known as the orthogonal projection of y onto uand, the vector z is known as thecomponentof y orthogonal to u. Typically, \(\widehat {\mathop{\rm y}\nolimits} \) is represented by \({{\mathop{\rm proj}\nolimits} _L}{\mathop{\rm y}\nolimits} \) and is called the orthogonal projection of y onto L.

\(\widehat {\mathop{\rm y}\nolimits} = {{\mathop{\rm proj}\nolimits} _L}{\mathop{\rm y}\nolimits} = \frac{{{\mathop{\rm y}\nolimits} \cdot {\mathop{\rm u}\nolimits} }}{{{\mathop{\rm u}\nolimits} \cdot {\mathop{\rm u}\nolimits} }}{\mathop{\rm u}\nolimits} \)

Compute \({\mathop{\rm y}\nolimits} \cdot {\mathop{\rm u}\nolimits} \) and \({\mathop{\rm u}\nolimits} \cdot {\mathop{\rm u}\nolimits} \) as shown below:

\(\begin{array}{c}{\mathop{\rm u}\nolimits} \cdot {\mathop{\rm y}\nolimits} = \left( {\begin{array}{*{20}{c}}7\\1\end{array}} \right) \cdot \left( {\begin{array}{*{20}{c}}2\\6\end{array}} \right)\\ = 20\\{\mathop{\rm u}\nolimits} \cdot {\mathop{\rm u}\nolimits} = \left( {\begin{array}{*{20}{c}}7\\1\end{array}} \right) \cdot \left( {\begin{array}{*{20}{c}}7\\1\end{array}} \right)\\ = 50\end{array}\)

Obtain the orthogonal projection of \({\mathop{\rm y}\nolimits} \) onto \({\mathop{\rm u}\nolimits} \) as shown below:

\(\begin{array}{c}\widehat {\mathop{\rm y}\nolimits} = \frac{{{\mathop{\rm y}\nolimits} \cdot {\mathop{\rm u}\nolimits} }}{{{\mathop{\rm u}\nolimits} \cdot {\mathop{\rm u}\nolimits} }}{\mathop{\rm u}\nolimits} \\ = \frac{{20}}{{50}}\left( {\begin{array}{*{20}{c}}7\\1\end{array}} \right)\\ = \frac{2}{5}\left( {\begin{array}{*{20}{c}}7\\1\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}{\frac{{14}}{5}}\\{\frac{2}{5}}\end{array}} \right)\end{array}\)

Thus, the orthogonal projection of \({\mathop{\rm y}\nolimits} \) onto \({\mathop{\rm u}\nolimits} \) is \(\widehat {\mathop{\rm y}\nolimits} = \left( {\begin{array}{*{20}{c}}{\frac{{14}}{5}}\\{\frac{2}{5}}\end{array}} \right)\).

Obtain the component of \({\mathop{\rm y}\nolimits} \) orthogonal to \({\mathop{\rm u}\nolimits} \) as shown below:

\(\begin{array}{c}{\mathop{\rm y}\nolimits} - \widehat {\mathop{\rm y}\nolimits} = \left( {\begin{array}{*{20}{c}}2\\6\end{array}} \right) - \left( {\begin{array}{*{20}{c}}{\frac{{14}}{5}}\\{\frac{2}{5}}\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}{2 - \frac{{14}}{5}}\\{6 - \frac{2}{5}}\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}{\frac{{ - 4}}{5}}\\{\frac{{28}}{5}}\end{array}} \right)\end{array}\)

Therefore, the component of \({\mathop{\rm y}\nolimits} \) orthogonal to \({\mathop{\rm u}\nolimits} \) is \({\mathop{\rm y}\nolimits} - \widehat {\mathop{\rm y}\nolimits} = \left( {\begin{array}{*{20}{c}}{\frac{{ - 4}}{5}}\\{\frac{{28}}{5}}\end{array}} \right)\).

Write \({\mathop{\rm y}\nolimits} \) as the sum of two orthogonal vectors as shown below:

\(\begin{array}{c}{\mathop{\rm y}\nolimits} = \widehat {\mathop{\rm y}\nolimits} + \left( {{\mathop{\rm y}\nolimits} - \widehat {\mathop{\rm y}\nolimits} } \right)\\ = \left( {\begin{array}{*{20}{c}}{\frac{{14}}{5}}\\{\frac{2}{5}}\end{array}} \right) + \left( {\begin{array}{*{20}{c}}{\frac{{ - 4}}{5}}\\{\frac{{28}}{5}}\end{array}} \right)\end{array}\)

Thus, the sum of two orthogonal vectors is \({\mathop{\rm y}\nolimits} = \left( {\begin{array}{*{20}{c}}{\frac{{14}}{5}}\\{\frac{2}{5}}\end{array}} \right) + \left( {\begin{array}{*{20}{c}}{\frac{{ - 4}}{5}}\\{\frac{{28}}{5}}\end{array}} \right)\).