Compute the orthogonal projection of 8 onto the line through 3 and the origin. The orthogonal projection is (Simplify vour answer 9.
Compute the orthogonal projection of 8 onto the line through 3 and the origin. The orthogonal projection is (Simplify vour answer 9.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![### Orthogonal Projection Problem and Solution
**Problem Statement:**
Compute the orthogonal projection of the vector
\[ \begin{bmatrix} 8 \\ 6 \end{bmatrix} \]
onto the line through
\[ \begin{bmatrix} -3 \\ 9 \end{bmatrix} \]
and the origin.
The orthogonal projection is:
\[ \boxed{} \]
(Simplify your answer)
**Explanation of the Problem:**
In this problem, you are asked to find the orthogonal projection of a given vector in 2D space onto a specific line. The line is defined by a vector and the origin. The line passes through the origin and the point given by the vector \(\begin{bmatrix} -3 \\ 9 \end{bmatrix}\).
### Steps to Compute the Orthogonal Projection:
1. **Understand the Vectors:**
- Original vector: \( \vec{v} = \begin{bmatrix} 8 \\ 6 \end{bmatrix} \)
- Line direction vector: \( \vec{a} = \begin{bmatrix} -3 \\ 9 \end{bmatrix} \)
2. **Formula for Orthogonal Projection:**
The orthogonal projection of \(\vec{v}\) onto \(\vec{a}\) is given by:
\[
\text{proj}_{\vec{a}} \vec{v} = \frac{\vec{v} \cdot \vec{a}}{\vec{a} \cdot \vec{a}} \vec{a}
\]
where \( \vec{v} \cdot \vec{a} \) is the dot product of \(\vec{v}\) and \(\vec{a}\), and \( \vec{a} \cdot \vec{a} \) is the dot product of \(\vec{a}\) with itself.
3. **Calculate the Dot Products:**
\[
\vec{v} \cdot \vec{a} = (8)(-3) + (6)(9) = -24 + 54 = 30
\]
\[
\vec{a} \cdot \vec{a} = (-3)^2 + 9^2 = 9 + 81 = 90
\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F780f9839-f737-4aba-91a2-6210989911b1%2F1b74f33f-7c25-42c3-a373-4157a771f44a%2F3n8trle_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Orthogonal Projection Problem and Solution
**Problem Statement:**
Compute the orthogonal projection of the vector
\[ \begin{bmatrix} 8 \\ 6 \end{bmatrix} \]
onto the line through
\[ \begin{bmatrix} -3 \\ 9 \end{bmatrix} \]
and the origin.
The orthogonal projection is:
\[ \boxed{} \]
(Simplify your answer)
**Explanation of the Problem:**
In this problem, you are asked to find the orthogonal projection of a given vector in 2D space onto a specific line. The line is defined by a vector and the origin. The line passes through the origin and the point given by the vector \(\begin{bmatrix} -3 \\ 9 \end{bmatrix}\).
### Steps to Compute the Orthogonal Projection:
1. **Understand the Vectors:**
- Original vector: \( \vec{v} = \begin{bmatrix} 8 \\ 6 \end{bmatrix} \)
- Line direction vector: \( \vec{a} = \begin{bmatrix} -3 \\ 9 \end{bmatrix} \)
2. **Formula for Orthogonal Projection:**
The orthogonal projection of \(\vec{v}\) onto \(\vec{a}\) is given by:
\[
\text{proj}_{\vec{a}} \vec{v} = \frac{\vec{v} \cdot \vec{a}}{\vec{a} \cdot \vec{a}} \vec{a}
\]
where \( \vec{v} \cdot \vec{a} \) is the dot product of \(\vec{v}\) and \(\vec{a}\), and \( \vec{a} \cdot \vec{a} \) is the dot product of \(\vec{a}\) with itself.
3. **Calculate the Dot Products:**
\[
\vec{v} \cdot \vec{a} = (8)(-3) + (6)(9) = -24 + 54 = 30
\]
\[
\vec{a} \cdot \vec{a} = (-3)^2 + 9^2 = 9 + 81 = 90
\
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