Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![**Expressing the Space Stretched by Vectors with Cartesian Equations**
**Vectors:**
\[ (2, -3, 1) \]
\[ (2, -3, 5) \]
**Problem Statement:**
Express the space stretched by the vectors \((2, -3, 1)\) and \((2, -3, 5)\) using Cartesian equations.
**Solution:**
To express the space stretched by the given vectors, we first need to identify the direction vectors and then use them to form a Cartesian equation.
Let \(\mathbf{v_1} = (2, -3, 1)\) and \(\mathbf{v_2} = (2, -3, 5)\).
1. **Compute the direction vectors:**
The direction vector \(\mathbf{d}\) is given by the difference between these two vectors:
\[
\mathbf{d} = \mathbf{v_2} - \mathbf{v_1} = (2, -3, 5) - (2, -3, 1)
\]
\[
\mathbf{d} = (2 - 2, -3 - (-3), 5 - 1) = (0, 0, 4)
\]
Therefore, \(\mathbf{d} = (0, 0, 4)\).
2. **Cartesian Equation:**
The space stretched by the vectors \(\mathbf{v_1}\) and \(\mathbf{v_2}\) is essentially a line. Since the direction vectors indicate the line stretches parallel to the z-axis (\((0, 0, 4)\)), we recognize it simplifies to:
\[
z = t \quad \text{(where \( t \in \mathbb{R} \))}
\]
The points lie on the line parallel to the \(z\)-axis passing through the points \((2, -3, 1)\) and \((2, -3, 5)\). The line is described in \( x \) and \( y \) as constant values \( x=2 \) and \( y=-3 \) respectively.
Thus, the Cartesian equations for the space stretched by these vectors are:
\[
\begin{cases}
x = 2, \\
y = -3, \\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc8241d13-b770-4dad-946c-6942db4056c7%2Fb9af1bf8-dc86-4008-939d-eb93f213f4d2%2Flg86ka_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Expressing the Space Stretched by Vectors with Cartesian Equations**
**Vectors:**
\[ (2, -3, 1) \]
\[ (2, -3, 5) \]
**Problem Statement:**
Express the space stretched by the vectors \((2, -3, 1)\) and \((2, -3, 5)\) using Cartesian equations.
**Solution:**
To express the space stretched by the given vectors, we first need to identify the direction vectors and then use them to form a Cartesian equation.
Let \(\mathbf{v_1} = (2, -3, 1)\) and \(\mathbf{v_2} = (2, -3, 5)\).
1. **Compute the direction vectors:**
The direction vector \(\mathbf{d}\) is given by the difference between these two vectors:
\[
\mathbf{d} = \mathbf{v_2} - \mathbf{v_1} = (2, -3, 5) - (2, -3, 1)
\]
\[
\mathbf{d} = (2 - 2, -3 - (-3), 5 - 1) = (0, 0, 4)
\]
Therefore, \(\mathbf{d} = (0, 0, 4)\).
2. **Cartesian Equation:**
The space stretched by the vectors \(\mathbf{v_1}\) and \(\mathbf{v_2}\) is essentially a line. Since the direction vectors indicate the line stretches parallel to the z-axis (\((0, 0, 4)\)), we recognize it simplifies to:
\[
z = t \quad \text{(where \( t \in \mathbb{R} \))}
\]
The points lie on the line parallel to the \(z\)-axis passing through the points \((2, -3, 1)\) and \((2, -3, 5)\). The line is described in \( x \) and \( y \) as constant values \( x=2 \) and \( y=-3 \) respectively.
Thus, the Cartesian equations for the space stretched by these vectors are:
\[
\begin{cases}
x = 2, \\
y = -3, \\
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