Find the equation of the line given by the intersection of the following two planes: T1:-2x + 4y- z+ 2=0; T2: 3r+2y – z = 1.
Find the equation of the line given by the intersection of the following two planes: T1:-2x + 4y- z+ 2=0; T2: 3r+2y – z = 1.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Please show me the how to solve
![**Problem:**
Find the equation of the line given by the intersection of the following two planes:
\( \pi_1: -2x + 4y - z + 2 = 0 \)
\( \pi_2: 3x + 2y - z = 1 \)
**Solution Overview:**
To find the equation of the line of intersection, we need to solve the system of linear equations represented by the two plane equations. This will give us a parametric form of the line, often expressed as:
\[ \mathbf{r}(t) = \mathbf{r_0} + t\mathbf{d} \]
where \( \mathbf{r_0} \) is a point on the line and \( \mathbf{d} \) is the direction vector.
**Detailed Explanation:**
1. **Setting up the System of Equations:**
- The plane equations can be rewritten as:
- \( -2x + 4y - z = -2 \)
- \( 3x + 2y - z = 1 \)
2. **Subtracting the Plane Equations:**
- By eliminating \( z \) through subtraction, we can simplify these equations:
\[
(3x + 2y - z) - (-2x + 4y - z) = 1 + 2
\]
\[
3x + 2y - z + 2x - 4y + z = 3
\]
\[
5x - 2y = 3
\]
3. **Solving for One Variable:**
- From \( 5x - 2y = 3 \), express one variable in terms of the other, e.g., solve for \( x \):
\[
x = \frac{3 + 2y}{5}
\]
4. **Substitute Back:**
- Substitute \( x = \frac{3 + 2y}{5} \) into one of the original plane equations to solve for \( z \).
5. **Parameterization:**
- Assign a parameter, \( t \), for one of the free variables, obtaining the parametric equations of the line:
\[
x = f(t), \quad y = g(t), \quad z = h(t)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbb7f1ce8-2552-49b5-ba1d-534a58de11ae%2F48d3f338-ad89-47a9-beee-160ba6420e77%2Fs8gk56_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem:**
Find the equation of the line given by the intersection of the following two planes:
\( \pi_1: -2x + 4y - z + 2 = 0 \)
\( \pi_2: 3x + 2y - z = 1 \)
**Solution Overview:**
To find the equation of the line of intersection, we need to solve the system of linear equations represented by the two plane equations. This will give us a parametric form of the line, often expressed as:
\[ \mathbf{r}(t) = \mathbf{r_0} + t\mathbf{d} \]
where \( \mathbf{r_0} \) is a point on the line and \( \mathbf{d} \) is the direction vector.
**Detailed Explanation:**
1. **Setting up the System of Equations:**
- The plane equations can be rewritten as:
- \( -2x + 4y - z = -2 \)
- \( 3x + 2y - z = 1 \)
2. **Subtracting the Plane Equations:**
- By eliminating \( z \) through subtraction, we can simplify these equations:
\[
(3x + 2y - z) - (-2x + 4y - z) = 1 + 2
\]
\[
3x + 2y - z + 2x - 4y + z = 3
\]
\[
5x - 2y = 3
\]
3. **Solving for One Variable:**
- From \( 5x - 2y = 3 \), express one variable in terms of the other, e.g., solve for \( x \):
\[
x = \frac{3 + 2y}{5}
\]
4. **Substitute Back:**
- Substitute \( x = \frac{3 + 2y}{5} \) into one of the original plane equations to solve for \( z \).
5. **Parameterization:**
- Assign a parameter, \( t \), for one of the free variables, obtaining the parametric equations of the line:
\[
x = f(t), \quad y = g(t), \quad z = h(t)
Expert Solution
Step 1
.jpg)
Step by step
Solved in 3 steps with 3 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
