20. Two lines F₁ = (1, 1,0) + t(1,-1, 2) and F2 = (2,0, 2) + (-1,1,0). (a) Find the point at which the two lines intersect. (b) Find an equation of the plane that contains these lines.
20. Two lines F₁ = (1, 1,0) + t(1,-1, 2) and F2 = (2,0, 2) + (-1,1,0). (a) Find the point at which the two lines intersect. (b) Find an equation of the plane that contains these lines.
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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20 please
![### Problem Statement on Lines and Planes
**20.** Two lines \( \mathbf{r}_1 \) and \( \mathbf{r}_2 \) are given by the parametric equations:
\[ \mathbf{r}_1 = (1,1,0) + t(1,-1,2) \]
\[ \mathbf{r}_2 = (2,0,2) + t(-1,1,0) \]
**Tasks:**
(a) Find the point at which the two lines intersect.
(b) Find an equation of the plane that contains these lines.
To approach this problem, we need to solve for the parameter \( t \) that makes the lines intersect and then use the result to find the plane.
For part (a), set the equations of \( \mathbf{r}_1 \) and \( \mathbf{r}_2 \) equal to find the intersection point.
For part (b), use the point found in (a) and the direction vectors of the two lines to determine the equation of the plane.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdb24b717-8ee5-4ca9-8061-74ddb7e91c1a%2F196ee603-fbf4-467d-bc63-744d98272685%2F6vxhlxn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement on Lines and Planes
**20.** Two lines \( \mathbf{r}_1 \) and \( \mathbf{r}_2 \) are given by the parametric equations:
\[ \mathbf{r}_1 = (1,1,0) + t(1,-1,2) \]
\[ \mathbf{r}_2 = (2,0,2) + t(-1,1,0) \]
**Tasks:**
(a) Find the point at which the two lines intersect.
(b) Find an equation of the plane that contains these lines.
To approach this problem, we need to solve for the parameter \( t \) that makes the lines intersect and then use the result to find the plane.
For part (a), set the equations of \( \mathbf{r}_1 \) and \( \mathbf{r}_2 \) equal to find the intersection point.
For part (b), use the point found in (a) and the direction vectors of the two lines to determine the equation of the plane.
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