Compute the orthogonal projection of v= 7 onto the line through 4 2 and the origin.
Compute the orthogonal projection of v= 7 onto the line through 4 2 and the origin.
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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![### Orthogonal Projection onto a Line
#### Problem Statement
Compute the orthogonal projection of \( \mathbf{v} = \begin{bmatrix} 1 \\ 7 \end{bmatrix} \) onto the line through \( \begin{bmatrix} -4 \\ 2 \end{bmatrix} \) and the origin.
#### Explanation
In this problem, we are given a vector \(\mathbf{v}\) and asked to find its orthogonal projection onto a line passing through another vector and the origin.
**Vector \(\mathbf{v}\)**:
\[ \mathbf{v} = \begin{bmatrix} 1 \\ 7 \end{bmatrix} \]
**Point on the line**:
\[ \begin{bmatrix} -4 \\ 2 \end{bmatrix} \]
The formula for the orthogonal projection of a vector \(\mathbf{v}\) onto a vector \(\mathbf{u}\) is given by:
\[ \text{proj}_{\mathbf{u}} \mathbf{v} = \frac{\mathbf{v} \cdot \mathbf{u}}{\mathbf{u} \cdot \mathbf{u}} \mathbf{u} \]
where:
- \( \cdot \) denotes the dot product.
Let's proceed step by step to solve this.
##### Step 1: Calculate the dot product \(\mathbf{v} \cdot \mathbf{u}\)
Given, \(\mathbf{u} = \begin{bmatrix} -4 \\ 2 \end{bmatrix}\),
\[ \mathbf{u} \cdot \mathbf{v} = (-4 \times 1) + (2 \times 7) = -4 + 14 = 10 \]
##### Step 2: Calculate the dot product \(\mathbf{u} \cdot \mathbf{u}\)
\[ \mathbf{u} \cdot \mathbf{u} = (-4 \times -4) + (2 \times 2) = 16 + 4 = 20 \]
##### Step 3: Calculate the projection
\[ \text{proj}_{\mathbf{u}} \mathbf{v} = \frac{10}{20} \begin{bmatrix} -4 \\ 2 \end{bmatrix} = \frac{1}{2} \begin{bmatrix](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9baea4c0-9d9e-491a-8271-bc2a2bbd8b45%2F0e433bee-c7a7-45df-88bf-18365b2dd96c%2Fbgztdd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Orthogonal Projection onto a Line
#### Problem Statement
Compute the orthogonal projection of \( \mathbf{v} = \begin{bmatrix} 1 \\ 7 \end{bmatrix} \) onto the line through \( \begin{bmatrix} -4 \\ 2 \end{bmatrix} \) and the origin.
#### Explanation
In this problem, we are given a vector \(\mathbf{v}\) and asked to find its orthogonal projection onto a line passing through another vector and the origin.
**Vector \(\mathbf{v}\)**:
\[ \mathbf{v} = \begin{bmatrix} 1 \\ 7 \end{bmatrix} \]
**Point on the line**:
\[ \begin{bmatrix} -4 \\ 2 \end{bmatrix} \]
The formula for the orthogonal projection of a vector \(\mathbf{v}\) onto a vector \(\mathbf{u}\) is given by:
\[ \text{proj}_{\mathbf{u}} \mathbf{v} = \frac{\mathbf{v} \cdot \mathbf{u}}{\mathbf{u} \cdot \mathbf{u}} \mathbf{u} \]
where:
- \( \cdot \) denotes the dot product.
Let's proceed step by step to solve this.
##### Step 1: Calculate the dot product \(\mathbf{v} \cdot \mathbf{u}\)
Given, \(\mathbf{u} = \begin{bmatrix} -4 \\ 2 \end{bmatrix}\),
\[ \mathbf{u} \cdot \mathbf{v} = (-4 \times 1) + (2 \times 7) = -4 + 14 = 10 \]
##### Step 2: Calculate the dot product \(\mathbf{u} \cdot \mathbf{u}\)
\[ \mathbf{u} \cdot \mathbf{u} = (-4 \times -4) + (2 \times 2) = 16 + 4 = 20 \]
##### Step 3: Calculate the projection
\[ \text{proj}_{\mathbf{u}} \mathbf{v} = \frac{10}{20} \begin{bmatrix} -4 \\ 2 \end{bmatrix} = \frac{1}{2} \begin{bmatrix
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