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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.