Find the solution set of the following vector equation.
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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![**3. Find the solution set of the following vector equation.**
\[
x_1
\begin{pmatrix}
-2 \\
3 \\
-3
\end{pmatrix}
+
x_2
\begin{pmatrix}
6 \\
-9 \\
9
\end{pmatrix}
+
x_3
\begin{pmatrix}
-6 \\
9 \\
-9
\end{pmatrix}
+
x_4
\begin{pmatrix}
-2 \\
1 \\
-2
\end{pmatrix}
=
\begin{pmatrix}
4 \\
0 \\
3
\end{pmatrix}
\]
In this equation, we are given a combination of vector coefficients multiplied by variables \((x_1, x_2, x_3, x_4)\) summing up to a result vector on the right-hand side. The goal is to find out the values of \(x_1, x_2, x_3,\) and \(x_4\) that satisfy the vector equation.
**Explanation of Vectors:**
1. The first vector is scaled by \(x_1\):
\(
\begin{pmatrix}
-2 \\
3 \\
-3
\end{pmatrix}
\)
2. The second vector is scaled by \(x_2\):
\(
\begin{pmatrix}
6 \\
-9 \\
9
\end{pmatrix}
\)
3. The third vector is scaled by \(x_3\):
\(
\begin{pmatrix}
-6 \\
9 \\
-9
\end{pmatrix}
\)
4. The fourth vector is scaled by \(x_4\):
\(
\begin{pmatrix}
-2 \\
1 \\
-2
\end{pmatrix}
\)
The combination of these vectors, across the respective \(x\) coefficients, should be equal to the vector on the right-hand side:
\(
\begin{pmatrix}
4 \\
0 \\
3
\end{pmatrix}
\)
**Approach for Solving:**
To solve this, you will need to set up a system](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F09b3997c-2096-4573-b74b-055c69181c0d%2F2a2dbbd0-7f86-4d6a-a9e7-da461728c346%2F09h7e_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**3. Find the solution set of the following vector equation.**
\[
x_1
\begin{pmatrix}
-2 \\
3 \\
-3
\end{pmatrix}
+
x_2
\begin{pmatrix}
6 \\
-9 \\
9
\end{pmatrix}
+
x_3
\begin{pmatrix}
-6 \\
9 \\
-9
\end{pmatrix}
+
x_4
\begin{pmatrix}
-2 \\
1 \\
-2
\end{pmatrix}
=
\begin{pmatrix}
4 \\
0 \\
3
\end{pmatrix}
\]
In this equation, we are given a combination of vector coefficients multiplied by variables \((x_1, x_2, x_3, x_4)\) summing up to a result vector on the right-hand side. The goal is to find out the values of \(x_1, x_2, x_3,\) and \(x_4\) that satisfy the vector equation.
**Explanation of Vectors:**
1. The first vector is scaled by \(x_1\):
\(
\begin{pmatrix}
-2 \\
3 \\
-3
\end{pmatrix}
\)
2. The second vector is scaled by \(x_2\):
\(
\begin{pmatrix}
6 \\
-9 \\
9
\end{pmatrix}
\)
3. The third vector is scaled by \(x_3\):
\(
\begin{pmatrix}
-6 \\
9 \\
-9
\end{pmatrix}
\)
4. The fourth vector is scaled by \(x_4\):
\(
\begin{pmatrix}
-2 \\
1 \\
-2
\end{pmatrix}
\)
The combination of these vectors, across the respective \(x\) coefficients, should be equal to the vector on the right-hand side:
\(
\begin{pmatrix}
4 \\
0 \\
3
\end{pmatrix}
\)
**Approach for Solving:**
To solve this, you will need to set up a system
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