$$|\mathbf{v}| = \sqrt{0^2 + 1^2 + 2^2} = \sqrt{5}$$
$$\hat{\mathbf{v}} = \dfrac{\mathbf{v}}{|\mathbf{v}|} = \dfrac{(0,1,2)}{\sqrt{5}} = \left(0,\dfrac{1}{\sqrt{5}},\dfrac{2}{\sqrt{5}}\right)$$
$$\mathbf{u} \cdot \hat{\mathbf{v}} = (1,2,3) \cdot \left(0,\dfrac{1}{\sqrt{5}},\dfrac{2}{\sqrt{5}}\right) = 1(0)+2\left(\dfrac{1}{\sqrt{5}}\right)+3\left(\dfrac{2}{\sqrt{5}}\right) = \dfrac{2+3(2)}{\sqrt{5}} = \dfrac{8}{\sqrt{5}} \approx 3.578$$
$$(\mathbf{u} \cdot \hat{\mathbf{v}})\hat{\mathbf{v}} = \dfrac{8}{\sqrt{5}}\left(0,\dfrac{1}{\sqrt{5}},\dfrac{2}{\sqrt{5}}\right) = \left(0,\dfrac{8}{5},\dfrac{16}{5}\right) = (0,1.6,3.2)$$