A civil engineer at the World Bank is designing a sustainable pedestrian bridge that follows a linear path defined by the parametric equation \(\mathbf{r}(t) = egin{pmatrix} 1 \ -2 \end{pmatrix} + t egin{pmatrix} 3 \ 1 \end{pmatrix}\). They need to locate the point on this line closest to a proposed water source located at \(egin{pmatrix} 5 \ 4 \end{pmatrix}\). Find the coordinates of this closest point.

["Designing a Sustainable Pedestrian Bridge with Precision: Finding the Closest Point on a Linear Path", "In civil engineering, especially in sustainable infrastructure projects, precision in design is critical—particularly when integrating pedestrian safety and accessibility into development plans. A recent project at the World Bank illustrates this challenge: designing a pedestrian bridge that follows a rational linear path defined by a parametric equation. The bridge’s trajectory is modeled by:", "[\n\mathbf{r}(t) = \begin{pmatrix} 1 \ -2 \end{pmatrix} + t \begin{pmatrix} 3 \ 1 \end{pmatrix}\n]", "This vector equation describes a straight line passing through point (\mathbf{P}_0 = (1, -2)) with direction vector (\mathbf{d} = \begin{pmatrix} 3 \ 1 \end{pmatrix}). To enhance accessibility and environmental harmony, the project requires identifying the point on this line closest to a proposed water source located at (\mathbf{Q} = (5, 4)), ensuring minimal footprint and optimal connectivity.", "### Finding the Closest Point on a Line to a Given Point", "To determine the point on the parametric line (\mathbf{r}(t)) closest to (Q = (5, 4)), we use vector projection techniques. The closest point occurs when the vector from (\mathbf{r}(t)) to (\mathbf{Q}) is perpendicular to the direction vector (\mathbf{d}).", "Let (\mathbf{r}(t) = \begin{pmatrix} 1 + 3t \ -2 + t \end{pmatrix}). Define the vector from (\mathbf{r}(t)) to (\mathbf{Q}):", "[\n\mathbf{v}(t) = \begin{pmatrix} 5 - (1 + 3t) \ 4 - (-2 + t) \end{pmatrix} = \begin{pmatrix} 4 - 3t \ 6 - t \end{pmatrix}\n]", "For (\mathbf{v}(t)) to be perpendicular to (\mathbf{d} = \begin{pmatrix} 3 \ 1 \end{pmatrix}), their dot product must be zero:", "[\n\mathbf{v}(t) \cdot \mathbf{d} = 0\n]", "Compute the dot product:", "[\n(4 - 3t)(3) + (6 - t)(1) = 0 \\n12 - 9t + 6 - t = 0 \\n18 - 10t = 0 \\n10t = 18 \\nt = \frac{9}{5} = 1.8\n]", "### Substituting (t = \frac{9}{5}) into (\mathbf{r}(t))", "Now plug (t = \frac{9}{5}) back into the parametric equation:", "[\nx = 1 + 3 \cdot \frac{9}{5} = 1 + \frac{27}{5} = \frac{32}{5} = 6.4 \\ny = -2 + \frac{9}{5} = -\frac{10}{5} + \frac{9}{5} = -\frac{1}{5} = -0.2\n]", "Thus, the coordinates of the point on the bridge path closest to the water source are:", "[\n\left( \frac{32}{5},\ -\frac{1}{5} \right)\n]", "### Why This Matters in Sustainable Infrastructure", "Identifying the closest point on a planned pedestrian route to a key facility like a water source enables engineers to minimize construction impact, preserve green corridors, and maximize access. It supports sustainable design by ensuring infrastructure integrates seamlessly with natural and social landscapes, reducing environmental disruption while enhancing community connectivity.", "Conclusion", "Through mathematical precision using vector geometry, civil engineers at the World Bank can strategically align sustainable infrastructure with environmental and social needs. The closest point on the linear bridge path to the water source at ((5, 4)) is (\boxed{ \left( \frac{32}{5},\ -\frac{1}{5} \right) })—a key data point guiding responsible, effective bridge placement in urban planning."]