-1 onto the line through 4 -3 and the origin. Compute the orthogonal projection of The orthogonal projection is
-1 onto the line through 4 -3 and the origin. Compute the orthogonal projection of The orthogonal projection is
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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Question
![### Orthogonal Projection Problem
**Problem Statement:**
Compute the orthogonal projection of \(\begin{bmatrix} -1 \\ 4 \end{bmatrix}\) onto the line through \(\begin{bmatrix} -3 \\ 5 \end{bmatrix}\) and the origin.
**Solution:**
The orthogonal projection is \(\begin{bmatrix} \boxed{} \\ \boxed{} \end{bmatrix}\).
Enter your answer in the edit fields and then click "Check Answer."
**Note:**
For computing the orthogonal projection of a vector \(\mathbf{a}\) onto another vector \(\mathbf{b}\), use the formula:
\[
\text{proj}_{\mathbf{b}} \mathbf{a} = \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{b} \cdot \mathbf{b}} \mathbf{b}
\]
where \(\mathbf{a} \cdot \mathbf{b}\) is the dot product of vectors \(\mathbf{a}\) and \(\mathbf{b}\).
### Instructions:
1. Calculate the dot product of the vectors.
2. Find the magnitude squared of vector \(\mathbf{b}\).
3. Use the projection formula to find the result.
4. Fill in the boxes provided with the correct values.
**Diagram and Explanation:**
(In this case, there are no diagrams provided. An explanation of steps and formulae is included above.)
After computing, enter your answers and click "Check Answer" to verify.
### Example Calculation:
Given:
\(\mathbf{a} = \begin{bmatrix} -1 \\ 4 \end{bmatrix}, \mathbf{b} = \begin{bmatrix} -3 \\ 5 \end{bmatrix}\)
1. Compute the dot product \(\mathbf{a} \cdot \mathbf{b}\):
\[
\mathbf{a} \cdot \mathbf{b} = (-1)(-3) + (4)(5) = 3 + 20 = 23
\]
2. Compute \(\mathbf{b} \cdot \mathbf{b}\):
\[
\mathbf{b} \cdot \mathbf{b} = (-3)^2 + 5^2 = 9 + 25 = 34
\]
3. Use the projection](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F05b5783f-5507-4009-950e-cf9b7485abe1%2F1723bb22-aea6-4891-b68b-ef7a11ba91be%2F8fjt9jn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Orthogonal Projection Problem
**Problem Statement:**
Compute the orthogonal projection of \(\begin{bmatrix} -1 \\ 4 \end{bmatrix}\) onto the line through \(\begin{bmatrix} -3 \\ 5 \end{bmatrix}\) and the origin.
**Solution:**
The orthogonal projection is \(\begin{bmatrix} \boxed{} \\ \boxed{} \end{bmatrix}\).
Enter your answer in the edit fields and then click "Check Answer."
**Note:**
For computing the orthogonal projection of a vector \(\mathbf{a}\) onto another vector \(\mathbf{b}\), use the formula:
\[
\text{proj}_{\mathbf{b}} \mathbf{a} = \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{b} \cdot \mathbf{b}} \mathbf{b}
\]
where \(\mathbf{a} \cdot \mathbf{b}\) is the dot product of vectors \(\mathbf{a}\) and \(\mathbf{b}\).
### Instructions:
1. Calculate the dot product of the vectors.
2. Find the magnitude squared of vector \(\mathbf{b}\).
3. Use the projection formula to find the result.
4. Fill in the boxes provided with the correct values.
**Diagram and Explanation:**
(In this case, there are no diagrams provided. An explanation of steps and formulae is included above.)
After computing, enter your answers and click "Check Answer" to verify.
### Example Calculation:
Given:
\(\mathbf{a} = \begin{bmatrix} -1 \\ 4 \end{bmatrix}, \mathbf{b} = \begin{bmatrix} -3 \\ 5 \end{bmatrix}\)
1. Compute the dot product \(\mathbf{a} \cdot \mathbf{b}\):
\[
\mathbf{a} \cdot \mathbf{b} = (-1)(-3) + (4)(5) = 3 + 20 = 23
\]
2. Compute \(\mathbf{b} \cdot \mathbf{b}\):
\[
\mathbf{b} \cdot \mathbf{b} = (-3)^2 + 5^2 = 9 + 25 = 34
\]
3. Use the projection
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