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Consider the curve r(t) = (t²- 4t, -3t+1, -2t² + 5t + 4). This is called a planar curve, which means it lies entirely n a single plane. Find an equation for the plane on which this curve lies.

Consider the curve r(t) = (t²- 4t, -3t+1, -2t² + 5t + 4). This is called a planar curve, which means it lies entirely n a single plane. Find an equation for the plane on which this curve lies.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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can you do this step by step using only math and not words so I can understand it better.

**Title: Understanding Planar Curves and Their Planes** **Topic: Planar Curves** Consider the curve \(\mathbf{r}(t) = \langle t^2 - 4t, -3t + 1, -2t^2 + 5t + 4 \rangle\). This is called a **planar curve**, which means it lies entirely on a single plane. **Problem Statement:** *Find an equation for the plane on which this curve lies.* In this task, you are given a vector function \(\mathbf{r}(t)\) that represents a curve in 3-dimensional space. Your objective is to determine an equation for the plane that contains this curve. **Steps to Solve:** 1. **Identify the Parameterization:** The provided vector function is \(\mathbf{r}(t) = \langle t^2 - 4t, -3t + 1, -2t^2 + 5t + 4 \rangle\). 2. **Isolate Components:** Break down the vector function into its component functions: - \(x(t) = t^2 - 4t\) - \(y(t) = -3t + 1\) - \(z(t) = -2t^2 + 5t + 4\) 3. **Determine Relationships:** Analyze the relationships between \(x\), \(y\), and \(z\) to find a common plane equation \(ax + by + cz = d\) that describes the plane containing the curve. By carefully studying and manipulating these component functions, we can derive an equation for the plane. This involves finding two tangent vectors to the curve, taking their cross product to get a plane normal vector, and then using a point on the curve to find the constant \(d\). **Note:** A more detailed step-by-step solution for finding the plane equation would involve specific algebraic manipulations and calculations, which can be included in further sections or examples. **Graphical Explanation:** If a graph or diagram were provided, it would illustrate the curve lying within a specific plane in the 3D coordinate system, clearly showing how the curve doesn't deviate from this plane. **Conclusion:** This exercise demonstrates how to identify and work with planar curves in the context of vector functions and multi-dimensional geometry. Finding the
Transcribed Image Text:**Title: Understanding Planar Curves and Their Planes** **Topic: Planar Curves** Consider the curve \(\mathbf{r}(t) = \langle t^2 - 4t, -3t + 1, -2t^2 + 5t + 4 \rangle\). This is called a **planar curve**, which means it lies entirely on a single plane. **Problem Statement:** *Find an equation for the plane on which this curve lies.* In this task, you are given a vector function \(\mathbf{r}(t)\) that represents a curve in 3-dimensional space. Your objective is to determine an equation for the plane that contains this curve. **Steps to Solve:** 1. **Identify the Parameterization:** The provided vector function is \(\mathbf{r}(t) = \langle t^2 - 4t, -3t + 1, -2t^2 + 5t + 4 \rangle\). 2. **Isolate Components:** Break down the vector function into its component functions: - \(x(t) = t^2 - 4t\) - \(y(t) = -3t + 1\) - \(z(t) = -2t^2 + 5t + 4\) 3. **Determine Relationships:** Analyze the relationships between \(x\), \(y\), and \(z\) to find a common plane equation \(ax + by + cz = d\) that describes the plane containing the curve. By carefully studying and manipulating these component functions, we can derive an equation for the plane. This involves finding two tangent vectors to the curve, taking their cross product to get a plane normal vector, and then using a point on the curve to find the constant \(d\). **Note:** A more detailed step-by-step solution for finding the plane equation would involve specific algebraic manipulations and calculations, which can be included in further sections or examples. **Graphical Explanation:** If a graph or diagram were provided, it would illustrate the curve lying within a specific plane in the 3D coordinate system, clearly showing how the curve doesn't deviate from this plane. **Conclusion:** This exercise demonstrates how to identify and work with planar curves in the context of vector functions and multi-dimensional geometry. Finding the
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