\mathbf{w} = \mathbf{w}_\parallel + \mathbf{w}_\perp, \quad \mathbf{w}_\parallel = \frac{\mathbf{v} \cdot \mathbf{w}}{\|\mathbf{v}\|^2} \mathbf{v} = \frac{6}{9} \mathbf{v} = \frac{2}{3} \mathbf{v}.