Find $\text {pro}_{\mathbf{w}} \mathbf{V}$ Then decompose v into two vectors, $\mathbf{v}_{1}$ and $\mathbf{v}_{2},$ where $\mathbf{v}_{1}$ is parallel to $\mathbf{w}$ and $\mathbf{v}_{2}$ is orthogonal to $\mathbf{w}.$
$$\mathbf{v}=\mathbf{i}+2 \mathbf{j}, \quad \mathbf{w}=3 \mathbf{i}+6 \mathbf{j}$$