Question: In a science experiment, a drone flies along the line $ \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 2 \\ -1 \end{pmatrix} + t \begin{pmatrix} 3 \\ 4 \end{pmatrix} $. Find the point on this line closest to $ (5, 7) $.
["Finding the Closest Point on a Line to a Given Point: A Science Experiment in 2D Geometry", "In modern scientific experiments involving motion tracking and path optimization, determining the closest point on a trajectory to a target location is crucial. This article details how to find the point on a parametric line closest to a given point—using the scientific method applied to vector geometry—specifically solving the example: A drone flies along the line ( \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} 2 \ -1 \end{pmatrix} + t \begin{pmatrix} 3 \ 4 \end{pmatrix} ); find the point on this line closest to ( (5, 7) ).", "---", "### Understanding the Problem", "In vector form, the drone’s path is defined by:\n[\n\mathbf{r}(t) = \begin{pmatrix} 2 \ -1 \end{pmatrix} + t \begin{pmatrix} 3 \ 4 \end{pmatrix}\n]\nWe are to find the point on this line closest to the point ( P = (5, 7) ).", "Geometrically, this closest point minimizes the Euclidean distance between ( P ) and any point on the line. This optimal point occurs where the vector from ( P ) to the line is perpendicular to the direction vector of the line.", "---", "### Step 1: Define Points and Vectors", "Let:\n- Point on the line at parameter ( t ):\n[\n\mathbf{r}(t) = \begin{pmatrix} 2 + 3t \ -1 + 4t \end{pmatrix}\n]\n- Target point: ( P = (5, 7) )", "The vector from the point on the line ( \mathbf{r}(t) ) to ( P ) is:\n[\n\mathbf{v}(t) = \begin{pmatrix} 5 - (2 + 3t) \ 7 - (-1 + 4t) \end{pmatrix} = \begin{pmatrix} 3 - 3t \ 8 - 4t \end{pmatrix}\n]", "The direction vector of the line is:\n[\n\mathbf{d} = \begin{pmatrix} 3 \ 4 \end{pmatrix}\n]", "---", "### Step 2: Enforce Perpendicularity", "For ( \mathbf{v}(t) ) to be perpendicular to ( \mathbf{d} ), their dot product must be zero:\n[\n\mathbf{v}(t) \cdot \mathbf{d} = 0\n]", "Compute the dot product:\n[\n(3 - 3t)(3) + (8 - 4t)(4) = 0\n]\n[\n9 - 9t + 32 - 16t = 0\n]\n[\n41 - 25t = 0\n]\n[\n25t = 41 \quad \Rightarrow \quad t = \frac{41}{25}\n]", "---", "### Step 3: Compute the Closest Point", "Substitute ( t = \frac{41}{25} ) into ( \mathbf{r}(t) ):\n[\nx = 2 + 3 \cdot \frac{41}{25} = 2 + \frac{123}{25} = \frac{50}{25} + \frac{123}{25} = \frac{173}{25}\n]\n[\ny = -1 + 4 \cdot \frac{41}{25} = -1 + \frac{164}{25} = \frac{-25}{25} + \frac{164}{25} = \frac{139}{25}\n]", "---", "### Final Answer", "The point on the drone’s path closest to ( (5, 7) ) is:\n[\n\left( \frac{173}{25},\ \frac{139}{25} \right)\n]", "---", "### Scientific Significance", "This method combines vector algebra and geometry to solve real-world problems in robotics, drone navigation, and physics simulations. By minimizing distance using perpendicularity, scientists and engineers ensure precise tracking and efficient path planning—key tools in experimental design and optimization under controlled conditions.", "Whether verifying theoretical models or calibrating real instruments, understanding the closest point on a line empowers accurate predictions and improved performance in dynamic systems.", "---", "Keywords: drone flight line, closest point geometry, vector optimization, parametric line, Euclidean distance minimization, vector dot product, line parametric equation, geometric algorithm, scientific experiment, 2D motion tracking."]
