Problem 78 - Calculus Early Transcendentals

Question

Describe the set of all points at which all three planes \(x+2 y+2 z=3, y+4 z=6,\) and \(x+2 y+8 z=9\) intersect.

Step-by-Step Solution

Answer

1) x + 2y + 2z = 3 2) y + 4z = 6 3) x + 2y + 8z = 9

Step 1: Rewrite the equations of the planes in matrix form

To make solving the system of linear equations easier, we can represent the planes' equations as a matrix. $$ \begin{pmatrix} 1 & 2 & 2 \\ 0 & 1 & 4 \\ 1 & 2 & 8 \\ \end{pmatrix} \begin{pmatrix} x \\ y \\ z \\ \end{pmatrix} = \begin{pmatrix} 3 \\ 6 \\ 9 \\ \end{pmatrix} $$

Step 2: Perform Gaussian elimination

To find the point of intersection, we perform Gaussian elimination on the matrix to get it into row-echelon form. Subtract the first row from the third row: $$ \begin{pmatrix} 1 & 2 & 2 \\ 0 & 1 & 4 \\ 0 & 0 & 6 \\ \end{pmatrix} \begin{pmatrix} x \\ y \\ z \\ \end{pmatrix} = \begin{pmatrix} 3 \\ 6 \\ 6 \\ \end{pmatrix} $$

Step 3: Perform back-substitution

Now that we have the matrix in row-echelon form, we can perform back-substitution to find the intersection point: Divide the third row by 6: $$z=\frac{6}{6}=1$$ Substitute z into the second row: $$y+4(1)=6 \Rightarrow y=6-4=2$$ Finally, substitute y and z into the first row: $$x+2(2)+2(1)=3 \Rightarrow x=3-4-2=-3$$

Step 4: Write the intersection point

The intersection point of all three planes is the solution to the system of linear equations, which is: Intersection point: \((-3, 2, 1)\)

Key Concepts

Gaussian Elimination

Gaussian Elimination is a systematic method used to solve systems of linear equations. It transforms a given set of simultaneous equations into simpler ones, making it easier to find the solutions. The goal is to reach a row-echelon form where the solution can be found through back-substitution.

Here's how you typically perform Gaussian Elimination:

• Step 1: Forward Elimination - Use row operations to create zeros below the pivot positions in the columns of the matrix.
• Step 2: Row Echelon Form - Your matrix will look like a staircase, with each leading entry in a subsequent row to the right of the one above it.
• Step 3: Back-Substitution - Begin with the last row to find the value of the corresponding variable and substitute back into the previous equations to find the other variables.

In the example exercise above, Gaussian Elimination was used to transform the system to a simpler form, culminating in finding the intersection point of three planes.

Matrix Representation

Matrix Representation is a powerful tool in linear algebra, allowing us to express a system of linear equations in a compact form. A matrix consists of rows and columns filled with coefficients of the equations involved. This provides a visual and computational advantage for solving the system.

This is how you represent equations in matrix form:

• Each equation corresponds to a row in the matrix, with coefficients of variables filling in the matrix's elements.
• The unknowns (variables) are placed into a column vector.
• The constants from the right-hand side of the equations form another column vector.

For the exercise, the plane equations were converted into a single matrix. This form aids in performing operations such as those in Gaussian Elimination, streamlining the solving process.

Intersection of Planes

The intersection of planes is an important geometrical concept when analyzing systems of equations in spaces. For our three-dimensional space, finding where three planes intersect involves solving a system of three equations.

When we have three planes,

• They might intersect at a single point.
• They could be parallel, leading to no intersection.
• They might intersect along a line if two planes overlap.

In the exercise example provided, the intersection point was found by solving the system representing the three planes. After using Gaussian Elimination, it was revealed that the planes intersect at the point \((-3, 2, 1)\). This point is where all three sets of equations are simultaneously satisfied, meaning all three planes share this common point in space.