Chapter 23: Chapter 23

Problem 585

Describe geometrically the solutions to $\mathrm{x}_{1}-2 \mathrm{x}_{2}+4 \mathrm{x}_{3}=1$ $3 \mathrm{x}_{1}+\mathrm{x}_{2}-\mathrm{x}_{3}=0$

Short Answer

Expert verified

The given system of linear equations represents two planes in 3-dimensional space, which are not parallel. They intersect in a line with the direction vector $\langle-2, 13, 7 \rangle$. The parametric equations of the intersection line are: $x = 1 - 4t + \frac{13t - 3}{3}$ $y = \frac{13t - 3}{6}$ $z = t$ These equations describe the geometric solution to the given system of linear equations.

See the step by step solution

Step by step solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Write the plane equations in the standard form

We'll write the given equation in the standard form (Ax + By + Cz = D) to make it easier to understand the geometric representations. The given equations are: 1. $x_1 - 2x_2 + 4x_3 = 1$ 2. $3x_1 + x_2 - x_3 = 0$ Now, let's rewrite these equations using x, y, and z instead of x1, x2, and x3: 1. $x - 2y + 4z = 1$ 2. $3x + y - z = 0$

Step 2: Identify the normal vectors of the planes

The normal vector to a plane can be found from the coefficients of the variables. For Ax + By + Cz = D, the normal vector  is orthogonal to the plane. Normal vector for the first plane: $<1, -2, 4>$ Normal vector for the second plane: $<3, 1, -1>$

Step 3: Check if the planes are parallel

Parallel planes never intersect. To check if the given planes are parallel, we need to check if their normal vectors are proportional. However, as we can see, the normal vectors are not proportional: $\frac{1}{3} \neq \frac{-2}{1} \neq \frac{4}{-1}$ Since the normal vectors are not proportional, the two planes are not parallel. Hence, they intersect in a line.

Step 4: Find the direction vector of the intersection line

Since we know the two planes intersect, their intersection line's direction vector can be found as the cross product of the normal vectors of the planes. Let's find the cross product of the two normal vectors: \[<1, -2, 4> \times <3, 1, -1> = <(-2)(-1) - (4)(1), (4)(3) - (1)(-1), (1)(1) - (-2)(3)>\] \[<2 - 4, 12 + 1, 1 + 6> = <-2, 13, 7>\] So, the direction vector of the intersection line is: $<-2, 13, 7>$

Step 5: Parametric equations of the line

To get the intersection line's parametric equations, rewrite the planes' equations with one variable as the parameter. Setting z as the parameter from the first equation: $t = z$ From equation 1: $x = 1 - 4t + 2y$ From equation 2: $3x + y - t = 0$ Now, let's substitute the first equation into the second equation: $3(1 - 4t + 2y) + y - t = 0$ Solve for y: $-12t + 6y + 3 - t = 0$ $-13t + 6y + 3 = 0$ $y = \frac{13t - 3}{6}$ Finally, we can rewrite x from equation 1 using y: $x = 1 - 4t + 2(\frac{13t - 3}{6})$ $x = 1 - 4t + \frac{13t - 3}{3}$ Thus, the parametric equations of the intersection line are: $x = 1 - 4t + \frac{13t - 3}{3}$ $y = \frac{13t - 3}{6}$ $z = t$ These equations provide the geometric description of the solutions to the given system of linear equations.

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Recommended explanations on Math Textbooks

Select your language

Chapter 23: Chapter 23

Describe geometrically the solutions to $\mathrm{x}_{1}-2 \mathrm{x}_{2}+4 \mathrm{x}_{3}=1$ \(3 \mathrm{x}_{1}+\mathrm{x}_{2}-\mathrm{x}_{3}=0\)

Short Answer

Step by step solution

Unlock all solutions

Step 1: Write the plane equations in the standard form

Step 2: Identify the normal vectors of the planes

Step 3: Check if the planes are parallel

Step 4: Find the direction vector of the intersection line

Step 5: Parametric equations of the line

Access millions of textbook solutions in one place

Most popular questions from this chapter

More chapters from the book ‘Linear Algebra’

Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 7

Chapter 8

Chapter 9

Chapter 10

Chapter 11

Chapter 12

Chapter 13

Chapter 14

Chapter 15

Chapter 16

Chapter 17

Chapter 18

Chapter 19

Chapter 20

Chapter 21

Chapter 22

Chapter 24

Join over 22 million students in learning with our Vaia App

Recommended explanations on Math Textbooks

Mechanics Maths

Probability and Statistics

Decision Maths

Pure Maths

Statistics

Geometry