Project the vector v=(1,1,1)\mathbf{v} = (1, 1, 1)v=(1,1,1) onto the subspace W=span{(1,1,0),(0,1,1)}W = \text{span}\{(1, 1, 0), (0, 1, 1)\}W=span{(1,1,0),(0,1,1)}.
projW(v)=(23,43,23)\text{proj}_W(\mathbf{v}) = \left(\frac{2}{3}, \frac{4}{3}, \frac{2}{3}\right)projW(v)=(32,34,32)
projW(v)=(1,1,1)\text{proj}_W(\mathbf{v}) = (1, 1, 1)projW(v)=(1,1,1)
projW(v)=(13,23,13)\text{proj}_W(\mathbf{v}) = \left(\frac{1}{3}, \frac{2}{3}, \frac{1}{3}\right)projW(v)=(31,32,31)
projW(v)=(23,23,23)\text{proj}_W(\mathbf{v}) = \left(\frac{2}{3}, \frac{2}{3}, \frac{2}{3}\right)projW(v)=(32,32,32)