# Let $y=\begin{bmatrix}2\\3\end{bmatrix} and u=\begin{bmatrix}4\\-7\end{bmatrix}$Write y as the sum of two

Let $y=\begin{bmatrix}2\\3\end{bmatrix}\ and\ u=\begin{bmatrix}4\\-7\end{bmatrix}$
Write y as the sum of two orthogonal vectors, one in Span{u} and one orthogonal to u.

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Muspee
Formula for orthogonal projection:
$$\displaystyle\hat{{{y}}}={\frac{{{y}\cdot{u}}}{{{u}\cdot{u}}}}{u}$$
$$\displaystyle{y}\cdot{u}={\left({2},{3}\right)}\cdot{\left({4},-{7}\right)}={8}-{21}=-{13}$$
$$\displaystyle{u}\cdot{u}={\left({4},-{7}\right)}\cdot{\left({4},-{7}\right)}={16}+{49}={65}$$
$\hat{y}=\frac{-13}{65}u=\frac{-1}{5}\begin{bmatrix}4\\-7 \end{bmatrix}=\begin{bmatrix}-4/5\\7/5\end{bmatrix}$
$$\displaystyle\hat{{{y}}}$$ is the part that is in span u. Now calculate other part:
$y-\hat{y}=\begin{bmatrix}2\\3\end{bmatrix}-\begin{bmatrix}-4/5\\7/5\end{bmatrix}=\begin{bmatrix}14/5\\8/5\end{bmatrix}$
$y=\begin{bmatrix}-4/5\\7/5\end{bmatrix}+\begin{bmatrix}14/5 \\8/5\end{bmatrix}$