(Solved): 1. (10 marks) Let \( W \) be a plane in \( \mathbb{R}^{3} \) given by \[ 3 x_{1}+2 x_{2}-5 x_{3}=0 ...

1. (10 marks) Let \( W \) be a plane in \( \mathbb{R}^{3} \) given by \[ 3 x_{1}+2 x_{2}-5 x_{3}=0 \] (a) Find the orthogonal complement of this plane. (b) Find the projection of a vector \( \mathbf{u}=\left[\begin{array}{c}1 \\ -1 \\ 4\end{array}\right] \) onto \( W \). (c) Find the distance from the vector \( \mathbf{u} \) to the plane \( W \). (d) What is the closest vector in \( W \) to the vector \( \mathbf{u} \) ?