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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.

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.

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 $v$.

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 $v$.