How do you verify that the residual vector is orthogonal to the projected vector in the example $y=(7,6)$ and $u=(2,1)$?

The residual vector is defined as $y$ minus its projection onto the line spanned by $v$. By the geometric definition of orthogonal projection, the difference between a vector and its projection onto a subspace is orthogonal to that subspace.

Conditions: $v$ is a nonzero vector.; The projection is orthogonal.

First, calculate the dot products $y \cdot u = 20$ and $u \cdot u = 5$. The scalar coefficient is $20/5 = 4$.

Conditions: $y = (7,6)$; $u = (2,1)$

The orthogonal projection of $y$ onto the line spanned by a nonzero vector $v$ is given by the formula $\frac{y \cdot v}{v \cdot v} v$. This formula scales the vector $v$ by the ratio of the dot product of $y$ and $v$ to the dot product of $v$ with itself.

Conditions: The vector $v$ must be nonzero.; The projection is onto the line spanned by $v$.

Calculate the projection $p$ of $y$ onto a line spanned by $v$. Calculate the residual $r = y - p$.

Conditions: $v$ is a nonzero vector defining the line.; $y$ is the vector to be decomposed.