Compute the orthogonal projection of = 3 onto the line L through -2 and the origin 2 proj, (v) =
Compute the orthogonal projection of = 3 onto the line L through -2 and the origin 2 proj, (v) =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![### Orthogonal Projection of a Vector onto a Line
**Problem Statement:**
Compute the orthogonal projection of \( \vec{v} = \begin{bmatrix} 8 \\ 3 \\ 9 \end{bmatrix} \) onto the line \( \mathcal{L} \) through \( \begin{bmatrix} 1 \\ -2 \\ 2 \end{bmatrix} \) and the origin.
**Mathematical Expression:**
The orthogonal projection of \( \vec{v} \) onto \( \mathcal{L} \) is given by:
\[ \text{proj}_{\mathcal{L}} (\vec{v}) = \begin{bmatrix} * \\ * \\ * \end{bmatrix} \]
where the asterisks represent the components of the projection vector.
### Explanation:
The orthogonal projection of vector \( \vec{v} \) onto a line \( \mathcal{L} \) can be computed using the formula:
\[ \text{proj}_{\vec{a}} \vec{v} = \frac{\vec{v} \cdot \vec{a}}{\vec{a} \cdot \vec{a}} \vec{a} \]
where \( \vec{a} \) is a direction vector of the line \( \mathcal{L} \), and \( \vec{v} \cdot \vec{a} \) denotes the dot product of \( \vec{v} \) and \( \vec{a} \).
Here, the line \( \mathcal{L} \) passes through \( \begin{bmatrix} 1 \\ -2 \\ 2 \end{bmatrix} \) and the origin \( \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} \), thus \( \vec{a} \) is \( \begin{bmatrix} 1 \\ -2 \\ 2 \end{bmatrix} \).
### Steps to Solve:
1. **Calculate the Dot Product**:
\[ \vec{v} \cdot \vec{a} = \begin{bmatrix} 8 \\ 3 \\ 9 \end{bmatrix} \cdot \begin{bmatrix} 1 \\ -2 \\ 2 \end{bmatrix}](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F30a42a85-c58f-45ac-a4af-faeed1a599e1%2F611d0ffa-293e-4c6a-894b-fddf7d5a2446%2Fzkh6n6d_processed.png&w=3840&q=75)
Transcribed Image Text:### Orthogonal Projection of a Vector onto a Line
**Problem Statement:**
Compute the orthogonal projection of \( \vec{v} = \begin{bmatrix} 8 \\ 3 \\ 9 \end{bmatrix} \) onto the line \( \mathcal{L} \) through \( \begin{bmatrix} 1 \\ -2 \\ 2 \end{bmatrix} \) and the origin.
**Mathematical Expression:**
The orthogonal projection of \( \vec{v} \) onto \( \mathcal{L} \) is given by:
\[ \text{proj}_{\mathcal{L}} (\vec{v}) = \begin{bmatrix} * \\ * \\ * \end{bmatrix} \]
where the asterisks represent the components of the projection vector.
### Explanation:
The orthogonal projection of vector \( \vec{v} \) onto a line \( \mathcal{L} \) can be computed using the formula:
\[ \text{proj}_{\vec{a}} \vec{v} = \frac{\vec{v} \cdot \vec{a}}{\vec{a} \cdot \vec{a}} \vec{a} \]
where \( \vec{a} \) is a direction vector of the line \( \mathcal{L} \), and \( \vec{v} \cdot \vec{a} \) denotes the dot product of \( \vec{v} \) and \( \vec{a} \).
Here, the line \( \mathcal{L} \) passes through \( \begin{bmatrix} 1 \\ -2 \\ 2 \end{bmatrix} \) and the origin \( \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} \), thus \( \vec{a} \) is \( \begin{bmatrix} 1 \\ -2 \\ 2 \end{bmatrix} \).
### Steps to Solve:
1. **Calculate the Dot Product**:
\[ \vec{v} \cdot \vec{a} = \begin{bmatrix} 8 \\ 3 \\ 9 \end{bmatrix} \cdot \begin{bmatrix} 1 \\ -2 \\ 2 \end{bmatrix}
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 3 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
