A hydrologist models contaminant spread using vectors $\mathbf{p} = \begin{pmatrix} 4 \\ -1 \\ 3 \end{pmatrix}$ and $\mathbf{q} = \begin{pmatrix} 2 \\ 0 \\ -1 \end{pmatrix}$. Find the projection of $\mathbf{p}$ onto $\mathbf{q}$.

["SEO Title: How Hydrologists Model Contaminant Spread Using Vector Projections: A Mathematical Approach with $\mathbf{p}$ and $\mathbf{q}$", "---", "Hydrologists play a crucial role in understanding and predicting how contaminants move through groundwater systems. A key mathematical tool in this modeling effort involves vector projections, which help quantify pollutant transport along preferred flow paths. In this article, we explore how vectors $\mathbf{p} = \begin{pmatrix} 4 \ -1 \ 3 \end{pmatrix}$ and $\mathbf{q} = \begin{pmatrix} 2 \ 0 \ -1 \end{pmatrix}$ are used to model contaminant dispersion, with a focus on computing the projection of $\mathbf{p}$ onto $\mathbf{q}$—a fundamental operation in hydrological vector analysis.", "---", "### Modeling Contaminant Spread in Groundwater", "Groundwater contaminants often spread along vectors representing the dominant direction and velocity of flow. By representing flow direction and contaminant concentration gradients as vectors, hydrologists can predict how pollutants migrate through porous media.", "Vectors $\mathbf{p}$ and $\mathbf{q}$ here represent competing flow and contaminant transport directions. Understanding the projection of $\mathbf{p}$ onto $\mathbf{q}$ provides insight into how much of $\mathbf{p}$ aligns with $\mathbf{q}$—critical for assessing whether pollutants follow major flow lines.", "---", "### The Projection of $\mathbf{p}$ onto $\mathbf{q}$", "In vector mathematics, the projection of vector $\mathbf{p}$ onto $\mathbf{q}$ gives the component of $\mathbf{p}$ that lies along $\mathbf{q}$. This projection vector $\ ext{proj}{\mathbf{q}} \mathbf{p}$ is computed using the formula:", "[\n\ ext{proj}}} \mathbf{p} = \left( \frac{\mathbf{p} \cdot \mathbf{q}}{\mathbf{q} \cdot \mathbf{q}} \right) \mathbf{q\n]", "---", "#### Step 1: Compute the Dot Product $\mathbf{p} \cdot \mathbf{q}$", "[\n\mathbf{p} \cdot \mathbf{q} = (4)(2) + (-1)(0) + (3)(-1) = 8 + 0 - 3 = 5\n]", "---", "#### Step 2: Compute $\mathbf{q} \cdot \mathbf{q}$ (the magnitude squared of $\mathbf{q}$)", "[\n\mathbf{q} \cdot \mathbf{q} = (2)^2 + (0)^2 + (-1)^2 = 4 + 0 + 1 = 5\n]", "---", "#### Step 3: Find the Scalar Projection Factor", "[\n\frac{\mathbf{p} \cdot \mathbf{q}}{\mathbf{q} \cdot \mathbf{q}} = \frac{5}{5} = 1\n]", "---", "#### Step 4: Multiply by $\mathbf{q}$ to get the projection vector", "[\n\ ext{proj}{\mathbf{q}} \mathbf{p} = 1 \cdot \begin{pmatrix} 2 \ 0 \ -1 \end{pmatrix} = \begin{pmatrix} 2 \ 0 \ -1 \end{pmatrix}\n]", "---", "### Interpretation", "The projection of $\mathbf{p}$ onto $\mathbf{q}$ is $\begin{pmatrix} 2 \ 0 \ -1 \end{pmatrix}$, which happens to be identical to $\mathbf{q}$ itself. This indicates that vector $\mathbf{p}$ completely aligns with the contaminant transport direction vector $\mathbf{q}$, suggesting a strong consistency between the modeled flow and pollutant spread direction.", "This alignment helps hydrologists confidently predict that contamination will propagate effectively along the $\mathbf{q}$-direction, informing remediation strategies and risk assessment.", "---", "Conclusion", "Understanding vector projections enables hydrologists to accurately model how contaminants travel through groundwater. By calculating $\ ext{proj}$, we quantify the overlap between flow vectors and pollutant transport pathways—key for sustainable water management and environmental protection.", "---", "}} \mathbf{pKeywords: hydrologist, contaminant spread, vector projection, $\mathbf{p}$ and $\mathbf{q}$ vectors, groundwater modeling, vector analysis, water resource science, contaminant transport, $\ ext{proj}{\mathbf{q}} \mathbf{p}$, $\mathbf{q} \cdot \mathbf{q}$, $\mathbf{p} \cdot \mathbf{q}$", "---", "Meta Description:\nDiscover how hydrologists use vector projections—like modeling the spread of contaminants using $\mathbf{p} = \begin{pmatrix} 4 \ -1 \ 3 \end{pmatrix}$ and $\mathbf{q} = \begin{pmatrix} 2 \ 0 \ -1 \end{pmatrix}$. Learn to compute $\ ext{proj}$ and understand its role in predicting pollutant migration through groundwater systems."]}} \mathbf{p