Consider the line through points A = (1, 1, 2) and B = (2, 3, 4), and let P be the foot of the perpendicular from Q = (3, -7, -1) to this line, as in the picture below:
Then \(\overrightarrow{AP}\) is the projection of \(\overrightarrow{AQ}\) onto a vector \(\mathbf{u} = \begin{pmatrix} u_1 \\ u_2 \\ u_3 \end{pmatrix}\)such that \(u_1 + u_2 + u_3 = 5.\) What is \(\mathbf{u} \)?
