1. **Problem Statement:** Find the projection of vector $\mathbf{v}$ onto vector $\mathbf{u}$, denoted as $\mathrm{proj}_{\mathbf{u}} \mathbf{v}$.\n\n2. **Formula:** The projection of $\mathbf{v}$ onto $\mathbf{u}$ is given by:\n$$\mathrm{proj}_{\mathbf{u}} \mathbf{v} = \frac{\mathbf{v} \cdot \mathbf{u}}{\mathbf{u} \cdot \mathbf{u}} \mathbf{u}$$\nwhere $\mathbf{v} \cdot \mathbf{u}$ is the dot product of $\mathbf{v}$ and $\mathbf{u}$.\n\n3. **Explanation:** The dot product $\mathbf{v} \cdot \mathbf{u}$ measures how much $\mathbf{v}$ points in the direction of $\mathbf{u}$. Dividing by $\mathbf{u} \cdot \mathbf{u}$ (which is $|\mathbf{u}|^2$) normalizes this measure relative to the length of $\mathbf{u}$. Multiplying by $\mathbf{u}$ scales the unit vector in the direction of $\mathbf{u}$ to the correct length.\n\n4. **Intermediate Steps:**\n- Compute $\mathbf{v} \cdot \mathbf{u}$.\n- Compute $\mathbf{u} \cdot \mathbf{u}$.\n- Divide the two results to get the scalar coefficient.\n- Multiply this scalar by $\mathbf{u}$ to get the projection vector.\n\n5. **Interpretation:** The vector $\mathrm{proj}_{\mathbf{u}} \mathbf{v}$ lies on the line defined by $\mathbf{u}$ and represents the shadow or component of $\mathbf{v}$ in the direction of $\mathbf{u}$.\n\nThis completes the calculation and understanding of the projection $\mathrm{proj}_{\mathbf{u}} \mathbf{v}$.