By first finding the projection onto V , find the projection of the given vector b Rm

QUESTION:

By first finding the projection onto \(V^{\perp}\), find the projection of the given vector \(\mathbf{b} \in \mathbb{R}^m\) onto the given hyperplane \(V \subset \mathbb{R}^m\).

a. \(V=\left(x_1+x_2+x_3=0\right) \subset \mathbb{R}^3, \mathbf{b}=(2,1,1)\)

b. \(V=\left(x_1+x_2+x_3=0\right) \subset \mathbb{R}^4, \mathbf{b}=(0,1,2,3)\)

c. \(V=\left\{x_1-x_2+x_3+2 x_4=0\right\} \subset \mathbb{R}^4, \mathbf{b}=(1,1,1,1)\)