Substitute the parametric expressions into the plane equation: $(2t + 1) + (-t + 4) + (3t - 2) = 15$.
["Substituting Parametric Expressions into a Plane Equation: A Step-by-Step Guide", "When modeling geometric objects like lines and planes in 3D space, parametric equations are often used to describe points along curves such as lines. Today, we explore an essential algebraic step: substituting parametric expressions into the general plane equation. This method is particularly useful in applications like computer graphics, physics simulations, and engineering design, where combining curve equations with plane constraints enables precise intersection and motion modeling.", "---", "### Understanding the Problem", "We begin with the parametric form of a line in 3D:", "$$\nt \ ext{ is a parameter } \quad \Rightarrow \quad \n\begin{cases}\nx = 2t + 1 \\ny = -t + 4 \\nz = 3t - 2\n\end{cases}\n$$", "Our goal is to substitute these expressions into the plane equation:", "$$\n(2t + 1) + (-t + 4) + (3t - 2) = 15\n$$", "This substitution effectively eliminates the parameter $ t $ and reduces the equation to a scalar identity involving only constants. Solving this reveals special values of $ t $, often corresponding to points of intersection or key geometric features.", "---", "### Step 1: Combine the Parametric Expressions", "Notice that the plane equation directly sums the three parametrized coordinates:", "- $ x = 2t + 1 $\n- $ y = -t + 4 $\n- $ z = 3t - 2 $", "So, the left-hand side of the equation becomes:", "$$\nx + y + z = (2t + 1) + (-t + 4) + (3t - 2)\n$$", "This matches exactly the expression we are given:", "$$\n(2t + 1) + (-t + 4) + (3t - 2) = 15\n$$", "---", "### Step 2: Simplify the Left-Hand Side", "Combine like terms involving $ t $ and constants:", "- Coefficient of $ t $: $ 2t - t + 3t = 4t $\n- Constant terms: $ 1 + 4 - 2 = 3 $", "So the equation reduces to:", "$$\n4t + 3 = 15\n$$", "---", "### Step 3: Solve for $ t $", "Subtract 3 from both sides:", "$$\n4t = 12\n$$", "Divide by 4:", "$$\nt = 3\n$$", "---", "### Step 4: Interpret the Result", "Having substituted the parametric expressions into the plane equation and solving for $ t $, we find that the point on the line where $ t = 3 $ lies on the plane:", "- $ x = 2(3) + 1 = 7 $\n- $ y = -3 + 4 = 1 $\n- $ z = 3(3) - 2 = 7 $", "Thus, the point $ (7, 1, 7) $ lies on both the parametric line and the plane $ x + y + z = 15 $.", "---", "### Why This Matters — Applications in Real-World Modeling", "Substituting parametric equations into plane equations enables precise computation in:", "- Computer Graphics: Finding intersections between parametric curves (e.g., camera paths, trajectories) and scene planes.\n- Robotics: Verifying if a robot’s path lies on or intersects useful spatial boundaries.\n- Physics Simulations: Locating specific points on motion paths relative to coordinate planes or reaction surfaces.", "This foundational technique bridges parametric description and geometric constraints, offering powerful tools for analysis and design.", "---", "### Final Summary", "To substitute parametric expressions into a plane equation:", "1. Replace $ x, y, z $ with their parametric forms.\n2. Simplify the resulting expression.\n3. Solve algebraically for the parameter $ t $.\n4. Interpret the result as a point or condition depending on application.", "By mastering this substitution step, you unlock deeper insight into how curves interact geometrically in three-dimensional space.", "---", "Keywords: parametric equations, plane equation, substituting into plane equation, 3D geometry, computer graphics, parametric to Cartesian conversion, point of intersection."]
