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Consider the lines r(t) = (2-3, +4--1) and r(s) (+4,- +3,3+6). A. Show that these two lines intersect and find the point of intersection. B. Find an equation for the plane that contains both lines.

Consider the lines r(t) = (2-3, +4--1) and r(s) (+4,- +3,3+6). A. Show that these two lines intersect and find the point of intersection. B. Find an equation for the plane that contains both lines.

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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### Intersection of Lines and Plane Equation Consider the lines \( \mathbf{r_1}(t) = \langle 2t - 3, t + 4, -t - 1 \rangle \) and \( \mathbf{r_2}(s) = \langle s + 6, s - 6, 3s + 6 \rangle \). #### Part A: **Objective:** Show that these two lines intersect and find the point of intersection. #### Part B: **Objective:** Find an equation for the plane that contains both lines. **Explanation:** 1. **Intersection of Lines:** - Determine the values of the parameters \(t\) and \(s\) where the coordinates of \(\mathbf{r_1}(t)\) and \(\mathbf{r_2}(s)\) are equal. - Solve the resulting system of equations to find the point of intersection. 2. **Equation of the Plane:** - Once the point of intersection is identified, use the direction vectors of \(\mathbf{r_1}(t)\) and \(\mathbf{r_2}(s)\) to form the normal vector to the plane. - Use the point-normal form of the equation of the plane to derive its equation. Please refer to relevant mathematical methods and algebraic tools to solve the above objectives.
Transcribed Image Text:### Intersection of Lines and Plane Equation Consider the lines \( \mathbf{r_1}(t) = \langle 2t - 3, t + 4, -t - 1 \rangle \) and \( \mathbf{r_2}(s) = \langle s + 6, s - 6, 3s + 6 \rangle \). #### Part A: **Objective:** Show that these two lines intersect and find the point of intersection. #### Part B: **Objective:** Find an equation for the plane that contains both lines. **Explanation:** 1. **Intersection of Lines:** - Determine the values of the parameters \(t\) and \(s\) where the coordinates of \(\mathbf{r_1}(t)\) and \(\mathbf{r_2}(s)\) are equal. - Solve the resulting system of equations to find the point of intersection. 2. **Equation of the Plane:** - Once the point of intersection is identified, use the direction vectors of \(\mathbf{r_1}(t)\) and \(\mathbf{r_2}(s)\) to form the normal vector to the plane. - Use the point-normal form of the equation of the plane to derive its equation. Please refer to relevant mathematical methods and algebraic tools to solve the above objectives.
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