The line in 5° Land (2.1,4 going through the erigin
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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Find the basis of the following subspace.
![**Title: Understanding Lines in Three-Dimensional Space**
**The Line in \( \mathbb{R}^3 \) Through the Origin and the Point (2,1,4)**
In three-dimensional space, \( \mathbb{R}^3 \), a line can be defined using vector equations. One such line is described by its passage through the origin, denoted as (0,0,0), and another point, here given as (2,1,4).
### Key Concepts:
- **Vector Representation**: A line in \( \mathbb{R}^3 \) can be represented by a parameterized equation using vectors. The general form of the equation is:
\[
\mathbf{r}(t) = \mathbf{r}_0 + t\mathbf{d}
\]
where \(\mathbf{r}_0\) is a position vector to a point on the line (here, the origin), \(\mathbf{d}\) is a direction vector (derived from the given points), and \(t\) is a scalar parameter.
- **Finding the Direction Vector**: Given the points (0,0,0) and (2,1,4), the direction vector \(\mathbf{d}\) is calculated as:
\[
\mathbf{d} = (2 - 0, 1 - 0, 4 - 0) = (2, 1, 4)
\]
- **Equation of the Line**: Substituting into the vector equation, the line through the origin and the point (2,1,4) can be described as:
\[
\mathbf{r}(t) = (0, 0, 0) + t(2, 1, 4)
\]
\[
\mathbf{r}(t) = (2t, t, 4t)
\]
This equation allows us to describe any point on the line by varying the parameter \(t\), illustrating how lines are constructed in a three-dimensional space. Understanding these concepts is crucial for applications in physics, engineering, and computer graphics where spatial reasoning is vital.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa824ef54-0ee2-4591-9058-ff8a94975446%2F547040ed-4d62-46e8-9501-e6a929fffdcb%2Fpijbl3_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Understanding Lines in Three-Dimensional Space**
**The Line in \( \mathbb{R}^3 \) Through the Origin and the Point (2,1,4)**
In three-dimensional space, \( \mathbb{R}^3 \), a line can be defined using vector equations. One such line is described by its passage through the origin, denoted as (0,0,0), and another point, here given as (2,1,4).
### Key Concepts:
- **Vector Representation**: A line in \( \mathbb{R}^3 \) can be represented by a parameterized equation using vectors. The general form of the equation is:
\[
\mathbf{r}(t) = \mathbf{r}_0 + t\mathbf{d}
\]
where \(\mathbf{r}_0\) is a position vector to a point on the line (here, the origin), \(\mathbf{d}\) is a direction vector (derived from the given points), and \(t\) is a scalar parameter.
- **Finding the Direction Vector**: Given the points (0,0,0) and (2,1,4), the direction vector \(\mathbf{d}\) is calculated as:
\[
\mathbf{d} = (2 - 0, 1 - 0, 4 - 0) = (2, 1, 4)
\]
- **Equation of the Line**: Substituting into the vector equation, the line through the origin and the point (2,1,4) can be described as:
\[
\mathbf{r}(t) = (0, 0, 0) + t(2, 1, 4)
\]
\[
\mathbf{r}(t) = (2t, t, 4t)
\]
This equation allows us to describe any point on the line by varying the parameter \(t\), illustrating how lines are constructed in a three-dimensional space. Understanding these concepts is crucial for applications in physics, engineering, and computer graphics where spatial reasoning is vital.
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