Given an \(n\times m\) matrix A and vectors \(b\in {\mathbb{R}}^ m\), \(p\in {\mathbb{R}}^ n\), the vector x with minimal distance to p is to be computed on the linear manifold \(\{A^ Tx=b\}\). The accumulation of rounding errors is studied for a modified Gram-Schmidt process, Householder, and Givens orthogonal factorization. If \(n\gg m\), then there is no substantial difference between Gram-Schmidt and the Householder method. On the other hand the Householder method is favorable if \(m\approx n\).